How to display get_categories selected category?

You need to save your selection and then call that selected value into your widget() method.

Here is an example of what I mean which displays the selected category name and description in the front end. You can use this widget as a base for your own widget. Just a note, this widget requires at least PHP 5.4

/**
 * Select_Category widget class
 *
 * Display a selected category
 *
 * @since 1.0.0
*/
class Select_Category extends WP_Widget 
{

    public function __construct() 
    {
        parent::__construct(
            'widget_selected_category', 
            _x( 'Selected Category Widget', 'Selected Category Widget' ), 
            [ 'description' => __( 'Display a selected category.' ) ] 
        );
        $this->alt_option_name="widget_selected_category";

        add_action( 'save_post', [$this, 'flush_widget_cache'] );
        add_action( 'deleted_post', [$this, 'flush_widget_cache'] );
        add_action( 'switch_theme', [$this, 'flush_widget_cache'] );
    }

    public function widget( $args, $instance ) 
    {
        $cache = [];
        if ( ! $this->is_preview() ) {
            $cache = wp_cache_get( 'widget_selected_cat', 'widget' );
        }

        if ( ! is_array( $cache ) ) {
            $cache = [];
        }

        if ( ! isset( $args['widget_id'] ) ) {
            $args['widget_id'] = $this->id;
        }

        if ( isset( $cache[ $args['widget_id'] ] ) ) {
            echo $cache[ $args['widget_id'] ];
            return;
        }

        ob_start();

        $title          = ( ! empty( $instance['title'] ) ) ? $instance['title'] : __( 'Category Posts' );
        /** This filter is documented in wp-includes/default-widgets.php */
        $title          = apply_filters( 'widget_title', $title, $instance, $this->id_base );
        if ( ! $number ) {
            $number = 5;
        }
        $cat_id       = $instance['cat_id'];


        echo $args['before_widget'];
        if ( $title ) {
            echo $args['before_title'] . $title . $args['after_title'];
        }   

        $select_cat_object = get_category( $cat_id );
        echo 'Your selected category is ' . $select_cat_object->name;
        echo "\n\t";
        echo $select_cat_object->description;

        echo $args['after_widget']; 

        if ( ! $this->is_preview() ) {
            $cache[ $args['widget_id'] ] = ob_get_flush();
            wp_cache_set( 'widget_selected_cat', $cache, 'widget' );
        } else {
            ob_end_flush();
        }
    }

    public function update( $new_instance, $old_instance ) 
    {
        $instance                   = $old_instance;
        $instance['title']          = strip_tags( $new_instance['title'] );
        $instance['cat_id']         = (int) $new_instance['cat_id'];

        $this->flush_widget_cache();

        $alloptions = wp_cache_get( 'alloptions', 'options' );
        if ( isset($alloptions['widget_selected_category']) )
            delete_option('widget_selected_category');

        return $instance;
    }

    public function flush_widget_cache() 
    {
        wp_cache_delete('widget_selected_cat', 'widget');
    }

    public function form( $instance ) 
    {

        $title  = isset( $instance['title'] ) ? esc_attr( $instance['title'] ) : '';
        $cat_id = isset( $instance['cat_id'] ) ? absint( $instance['cat_id'] ) : 1;
        ?>

        <p>
            <label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
            <input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo $title; ?>" />
        </p>

        <p>
            <label for="<?php echo $this->get_field_id('cat_id'); ?>"><?php _e( 'Category Name:' )?></label>
            <select id="<?php echo $this->get_field_id('cat_id'); ?>" name="<?php echo $this->get_field_name('cat_id'); ?>">
                <?php 
                $this->categories = get_categories();
                foreach ( $this->categories as $cat ) {
                    $selected = ( $cat->term_id == esc_attr( $cat_id ) ) ? ' selected = "selected" ' : '';
                    $option = '<option '.$selected .'value="' . $cat->term_id;
                    $option = $option .'">';
                    $option = $option .$cat->name;
                    $option = $option .'</option>';
                    echo $option;
                }
                ?>
            </select>
        </p>

    <?php
    }

}

add_action( 'widgets_init', function () 
{
    register_widget( 'Select_Category' );
});