How to show a widget for logged-in users only?

Your widget is an extension of the WP_Widget class, which defines the widget function. This is where your code should go.

class WPDev_156470_Widget extends WP_Widget {

    // other stuff

    public function widget( $args, $instance ) {

        if ( ! is_user_logged_in() ) {
            echo '';
            return;
        }

        // logged-in-user stuff
    }

    // other stuff

}

// EDIT

<?php

/**
 * Register the widget
 */
function wpspprofwid_load_widgets() {
    register_widget( 'WPS_PRO_Profile_Widget' );
}
add_action( 'widgets_init', 'wpspprofwid_load_widgets' );

/**
 * Profile Widget class.
 */
class WPS_PRO_Profile_Widget extends WP_Widget {

    /**
     * Widget setup.
     */
    public function __construct() {
        $widget_ops = array( 'description' => __('Add a custom Profile to your sidebar.') );
        parent::__construct( 'nav_menu1', __('WP Symposium PRO Profile'), $widget_ops );
    }



    // ********************** How to display Widget on Screen *******************************
    public function widget($args, $instance) {


        // THIS is where to ADD the code   
        if ( ! is_user_logged_in() ) {
            echo '';
            return;
        }


        // Set Title
        $title = apply_filters( 'widget_title', $instance['title'] );

        // Display before widget code
        echo $before_widget;

        //Display Title
        if ( ! empty( $title ) )
            echo $args['before_title'] . $title . $args['after_title'];

        /* ========================= Avatar image start ====================================*/

        if($instance['avtar_size']!='')
        {

            echo do_shortcode('[wps-avatar size="'.$instance['avtar_size'].'"]');
        }
        /* ========================= Avatar image end ====================================*/
        // Do Shortcodes

        echo "<br>"; // space
        echo do_shortcode('[wps-friends-add-button]');
        echo "<br>"; // space

        // ====================== Menu Display ==================================

        //Set Menu title
        $mtitle = apply_filters( 'Menu Title', $instance['mtitle'] );

        //Display Title
        if ( ! empty( $mtitle ) )
            echo $args['before_mtitle'] . "<h3>$mtitle</h3>" . $args['after_mtitle'];

        // Get menu
        $nav_menu = ! empty( $instance['nav_menu'] ) ? wp_get_nav_menu_object( $instance['nav_menu'] ) : false;

        if($nav_menu != '')
        {
            wp_nav_menu( array( 'fallback_cb' => '', 'menu' => $nav_menu ) );
        }

        /* ========================= Friends shortcode start ====================================*/

        if($instance['friends_shortcode']=='on')
        {
            echo "<h3>My Friends</h3>";
            echo do_shortcode('[wps-friends size="35" count=5]');
        }

        /* ========================= Friends Page URL start ==================================*/

        echo '<p><a href="'.$instance ['friends_url'].'">'.$instance['fptitle'].'</a></p>';

        /* ========================= First Shortcode section start ===================================*/
        //Set Shortcode 1 title
        $sc1title = apply_filters( 'Shortcode 1 Title', $instance['sc1title'] );

        //Display Shortcode 1 title
        if ( ! empty( $sc1title ) )
            echo $args['before_sc1title'] . "<h3>$sc1title</h3>" . $args['after_sc1title'];

        //Do Shortcode 1
        echo do_shortcode($instance['sc1']);
        //echo "<br>";

        /* ========================= Second Shortcode section start ===============================*/

        //Set Shortcode 2 title
        $sc2title = apply_filters( 'Shortcode 2 Title', $instance['sc2title'] );

        //Display Shortcode 2 title
        if ( ! empty( $sc2title ) )
            echo $args['before_sc1title'] . "<h3>$sc2title</h3>" . $args['after_sc1title'];

        //Do Shortcode 2
        echo do_shortcode($instance['sc2']);
        //echo "<br>";

        /* ========================= Third Shortcode section start =====================*/

        //Set Shortcode 3 title
        $sc3title = apply_filters( 'Shortcode 3 Title', $instance['sc3title'] );

        //Display Shortcode 3 title
        if ( ! empty( $sc3title ) )
            echo $args['before_sc1title'] . "<h3>$sc3title</h3>" . $args['after_sc1title'];

        //Do Shortcode 3
        echo do_shortcode($instance['sc3']);
        //echo "<br>";

        /* ========================= Fourth Shortcode section start ==========================*/

        //Set Shortcode 4 title
        $sc4title = apply_filters( 'Shortcode 4 Title', $instance['sc4title'] );

        //Display Shortcode 4 title
        if ( ! empty( $sc4title ) )
            echo $args['before_sc1title'] . "<h3>$sc4title</h3>" . $args['after_sc1title'];

        //Do Shortcode 4
        echo do_shortcode($instance['sc4']);
        //echo "<br>";

        /* ========================= Fifth Shortcode section start ==========================*/

        //Set Shortcode 5 title
        $sc5title = apply_filters( 'Shortcode 5 Title', $instance['sc5title'] );

        //Display Shortcode 5 title
        if ( ! empty( $sc5title ) )
            echo $args['before_sc1title'] . "<h3>$sc5title</h3>" . $args['after_sc1title'];

        //Do Shortcode 5
        echo do_shortcode($instance['sc5']);
        //echo "<br>";

        /* ========================= Sixth Shortcode section start =============================*/

        //Set Shortcode 6 title
        $sc6title = apply_filters( 'Shortcode 6 Title', $instance['sc6title'] );

        //Display Shortcode 6 title
        if ( ! empty( $sc6title ) )
            echo $args['before_sc1title'] . "<h3>$sc6title</h3>" . $args['after_sc1title'];

        //Do Shortcode 6
        echo do_shortcode($instance['sc6']);
        echo "<br>"
        echo $args['after_widget'];

        // Display after widget code
        echo $after_widget;
    }

    // Update the widget settings.
    public function update( $new_instance, $old_instance ) {
        $instance['title'] = strip_tags( stripslashes($new_instance['title']) );
        $instance['avtar_size'] =  strip_tags( stripslashes($new_instance['avtar_size']) );
        $instance['mtitle'] = strip_tags( stripslashes($new_instance['mtitle']) );
        $instance['nav_menu'] = (int) $new_instance['nav_menu'];
        $instance['friends_shortcode'] =  $new_instance['friends_shortcode'];
        $instance['fptitle'] = strip_tags( stripslashes($new_instance['fptitle']) );
        $instance['friends_url'] = strip_tags( stripslashes($new_instance['friends_url']) );
        $instance['sc1title'] = strip_tags( stripslashes($new_instance['sc1title']) );
        $instance['sc1'] = strip_tags( stripslashes($new_instance['sc1']) );
        $instance['sc2title'] = strip_tags( stripslashes($new_instance['sc2title']) );
        $instance['sc2'] = strip_tags( stripslashes($new_instance['sc2']) );
        $instance['sc3title'] = strip_tags( stripslashes($new_instance['sc3title']) );
        $instance['sc3'] = strip_tags( stripslashes($new_instance['sc3']) );
        $instance['sc4title'] = strip_tags( stripslashes($new_instance['sc4title']) );
        $instance['sc4'] = strip_tags( stripslashes($new_instance['sc4']) );
        $instance['sc5title'] = strip_tags( stripslashes($new_instance['sc5title']) );
        $instance['sc5'] = strip_tags( stripslashes($new_instance['sc5']) );
        $instance['sc6title'] = strip_tags( stripslashes($new_instance['sc6title']) );
        $instance['sc6'] = strip_tags( stripslashes($new_instance['sc6']) );
        return $instance;
    }

    // =============== Displays widget in Admin Panel

    public function form( $instance ) {
        $title = isset( $instance['title'] ) ? $instance['title'] : '';
        $avtar_size = isset( $instance['avtar_size'] ) ? $instance['avtar_size'] : '';
        $mtitle = isset( $instance['mtitle'] ) ? $instance['mtitle'] : '';
        $nav_menu = isset( $instance['nav_menu'] ) ? $instance['nav_menu'] : '';
        $friends_shortcode = isset( $instance['friends_shortcode'] ) ? $instance['friends_shortcode'] : '';
        $fptitle = isset( $instance['fptitle'] ) ? $instance['fptitle'] : '';
        $friends_url = isset( $instance['friends_url'] ) ? $instance['friends_url'] : '';
        $sc1title = isset( $instance['sc1title'] ) ? $instance['sc1title'] : '';
        $sc1 = isset( $instance['sc1'] ) ? $instance['sc1'] : '';
        $sc2title = isset( $instance['sc2title'] ) ? $instance['sc2title'] : '';
        $sc2 = isset( $instance['sc2'] ) ? $instance['sc2'] : '';
        $sc3title = isset( $instance['sc3title'] ) ? $instance['sc3title'] : '';
        $sc3 = isset( $instance['sc3'] ) ? $instance['sc3'] : '';
        $sc4title = isset( $instance['sc4title'] ) ? $instance['sc4title'] : '';
        $sc4 = isset( $instance['sc4'] ) ? $instance['sc4'] : '';
        $sc5title = isset( $instance['sc5title'] ) ? $instance['sc5title'] : '';
        $sc5 = isset( $instance['sc5'] ) ? $instance['sc5'] : '';
        $sc6title = isset( $instance['sc6title'] ) ? $instance['sc6title'] : '';
        $sc6 = isset( $instance['sc6'] ) ? $instance['sc6'] : '';

        // Get menus
        $menus = wp_get_nav_menus( array( 'orderby' => 'name' ) );

        // If no menus exists, direct the user to go and create some.
        if ( !$menus ) {
            echo '<p>'. sprintf( __('No menus have been created yet. <a href="%s">Create some</a>.'), admin_url('nav-menus.php') ) .'</p>';
            return;
        }
        // ============= Widget Selectio s and inputs
        ?>
        <p>
            <label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:', 'wpspprofwid' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('title'); ?>" name="<?php echo $this->get_field_name('title'); ?>" value="<?php echo $title; ?>" />
        </p>
        <p>
            <div>Select Avatar Size Below:</div>
            <label for="<?php echo $this->get_field_id( 'avtar_size' ); ?>"><?php _e( 'Avatar Size: ' ); ?></label>
            <select id="<?php echo $this->get_field_id( 'avtar_size' ); ?>" name="<?php echo $this->get_field_name('avtar_size'); ?>">
                <option value=""> - No Avatar - </option>
                <?php
                echo '<option value="200"'. selected( $avtar_size, '200', false ). '>200 x 200px</option>';
                echo '<option value="150"'. selected( $avtar_size, '150', false ). '>150 x 150px</option>';
                echo '<option value="100"'. selected( $avtar_size, '100', false ). '>100 x 100px</option>';
                ?>
            </select>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'mtitle' ); ?>"><?php _e( 'User Menu Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('mtitle'); ?>" name="<?php echo $this->get_field_name('mtitle'); ?>" value="<?php echo $mtitle; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'nav_menu' ); ?>"><?php _e( 'Set User Menu:' ); ?></label>
            <select id="<?php echo $this->get_field_id( 'nav_menu' ); ?>" name="<?php echo $this->get_field_name('nav_menu'); ?>">
                <option value="">Select Menu</option>
                <?php
                foreach ( $menus as $menu ) {
                    echo '<option value="' . $menu->term_id . '"'
                        . selected( $nav_menu, $menu->term_id, false )
                        . '>'. $menu->name . '</option>';
                }
                ?>
            </select>
        </p>
        <p>
            <div>Check Box to Show Friends</div>
            <label for="<?php echo $this->get_field_id( 'friends_shortcode' ); ?>"><?php _e( 'Show Friends: ' ); ?></label>
            <input class="checkbox" type="checkbox" <?php checked($friends_shortcode , 'on'); ?> id="<?php echo $this->get_field_id('friends_shortcode'); ?>" name="<?php echo $this->get_field_name('friends_shortcode'); ?>" />
        </p>
        <p>
            <div>Link for Friends Page - leave blank for no link</div>
            <label for="<?php echo $this->get_field_id( 'fptitle' ); ?>"><?php _e( 'Friend Page Label - ex. See All Friends' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('fptitle'); ?>" name="<?php echo $this->get_field_name('fptitle'); ?>" value="<?php echo $fptitle; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'friends_url' ); ?>"><?php _e( 'URL to Friends Page:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('friends_url'); ?>" name="<?php echo $this->get_field_name('friends_url'); ?>" value="<?php echo $friends_url; ?>" />
            <p>Add Additional Shortcodes Below:</p>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc1title' ); ?>"><?php _e( 'Shortcode 1 Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('sc1title'); ?>" name="<?php echo $this->get_field_name('sc1title'); ?>" value="<?php echo $sc1title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc1' ); ?>"><?php _e( 'Shortcode 1:' ); ?></label>
            <textarea name="<?php echo $this->get_field_name('sc1'); ?>" id="<?php echo $this->get_field_id('sc1'); ?>" class="widefat"><?php echo $sc1; ?></textarea>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc1title' ); ?>"><?php _e( 'Shortcode 2 Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('sc2title'); ?>" name="<?php echo $this->get_field_name('sc2title'); ?>" value="<?php echo $sc2title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc2' ); ?>"><?php _e( 'Shortcode 2:' ); ?></label>
            <textarea name="<?php echo $this->get_field_name('sc2'); ?>" id="<?php echo $this->get_field_id('sc2'); ?>" class="widefat"><?php echo $sc2; ?></textarea>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc3title' ); ?>"><?php _e( 'Shortcode 3 Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('sc3title'); ?>" name="<?php echo $this->get_field_name('sc3title'); ?>" value="<?php echo $sc3title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc3' ); ?>"><?php _e( 'Shortcode 3:' ); ?></label>
            <textarea name="<?php echo $this->get_field_name('sc3'); ?>" id="<?php echo $this->get_field_id('sc3'); ?>" class="widefat"><?php echo $sc3; ?></textarea>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc4title' ); ?>"><?php _e( 'Shortcode 4 Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('sc4title'); ?>" name="<?php echo $this->get_field_name('sc4title'); ?>" value="<?php echo $sc4title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc4' ); ?>"><?php _e( 'Shortcode 4:' ); ?></label>
            <textarea name="<?php echo $this->get_field_name('sc4'); ?>" id="<?php echo $this->get_field_id('sc4'); ?>" class="widefat"><?php echo $sc4; ?></textarea>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc5title' ); ?>"><?php _e( 'Shortcode 5 Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('sc5title'); ?>" name="<?php echo $this->get_field_name('sc5title'); ?>" value="<?php echo $sc5title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc5' ); ?>"><?php _e( 'Shortcode 5:' ); ?></label>
            <textarea name="<?php echo $this->get_field_name('sc5'); ?>" id="<?php echo $this->get_field_id('sc5'); ?>" class="widefat"><?php echo $sc5; ?></textarea>
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc6title' ); ?>"><?php _e( 'Shortcode 6 Title:' ); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('sc6title'); ?>" name="<?php echo $this->get_field_name('sc6title'); ?>" value="<?php echo $sc6title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id( 'sc6' ); ?>"><?php _e( 'Shortcode 6:' ); ?></label>
            <textarea name="<?php echo $this->get_field_name('sc6'); ?>" id="<?php echo $this->get_field_id('sc6'); ?>" class="widefat"><?php echo $sc6; ?></textarea>
        </p>
    <?php
    }
}
?>