Add URL to selected post in meta box

Super short version:

Replace $post->post_title with get_permalink($post->ID).

Short version:

Add a filter to your current Code, where you output the post_title.

$posts = get_posts(array('post_type'=> 'lesson', 'post_status'=> 'publish', 'suppress_filters' => false, 'posts_per_page'=>-1));
        //here you add the HTML of the dropdown you add something like

        echo '<p>Select the lesson: <select name="_dappcf_i_dropdown" class="widefat"  style="width:170px" >';
        foreach ($posts as $post) {
        echo '<option value="', $post->ID, '"'; 
        if  ($my_dropdown == $post->ID){echo ' selected="selected"';} 
        echo '>'.
        apply_filters('filter_list_cpt', $post->post_title, $post).
        '</option>';

        }
    echo '</select>';

Somewhere in your functions.php or plugin file:

function filter_list_cpt($title, $cpt) {
    return sprintf(__('%1$s [%2$s]', 'my_textdomain'), $title, get_permalink($cpt->ID));
}

add_filter('list_cpt', 'filter_list_cpt', 10, 2);

But why stop here? Let us improve the provided code and put it into a WordPress perspective.

Better approach

WordPress has its own helper functions to return a select of e.g. pages. So instead of just writing code, lets try to copy and rewrite this function to our needs. I call it: my_dropdown_post_type

function my_dropdown_post_type( $post_type="lesson", $args="" ) {
    $defaults = array(
        'depth' => 0, 'child_of' => 0,
        'selected' => 0, 'echo' => 1,
        'name' => "{$post_type}_id", 'id' => '',
        'class' => '',
        'show_option_none' => '', 'show_option_no_change' => '',
        'option_none_value' => '',
        'value_field' => 'ID',
        'post_type' => $post_type
    );

    $r = wp_parse_args( $args, $defaults );

    $cpt = get_posts( $r );
    $output="";

    if ( ! empty( $cpt ) ) {
        $class="";
        if ( ! empty( $r['class'] ) ) {
            $class = " class="" . esc_attr( $r["class'] ) . "'";
        }

        $output = "<select name="" . esc_attr( $r["name'] ) . "'" . $class . " id='" . esc_attr( $r['id'] ) . "'>\n";
        if ( $r['show_option_no_change'] ) {
            $output .= "\t<option value=\"-1\">" . $r['show_option_no_change'] . "</option>\n";
        }
        if ( $r['show_option_none'] ) {
            $output .= "\t<option value=\"" . esc_attr( $r['option_none_value'] ) . '">' . $r['show_option_none'] . "</option>\n";
        }
        $output .= walk_page_dropdown_tree( $cpt, $r['depth'], $r );
        $output .= "</select>\n";
    }

    $html = apply_filters( 'wp_dropdown_pages', $output, $r, $cpt );

    if ( $r['echo'] ) {
        echo $html;
    }
    return $html;
}

Then we apply our list_pages filter

function filter_list_cpt($title, $cpt) {
    if ($cpt->post_type == 'lesson')
        return sprintf(__('%1$s [%2$s]', 'my_textdomain'), $title, get_permalink($cpt->ID));
    return $title;
}

add_filter('list_pages', 'filter_list_cpt', 10, 2);

In the end we can call our my_dropdown_post_type function like any other basic WP function:

my_dropdown_post_type(
    'lesson', 
    array(
        'selected' => $my_dropdown, 
        'post_status'=> 'publish', 
        'suppress_filters' => false, 
        'posts_per_page'=>-1
    )
);