How do I select an image from Media Library in my plugin?

You should use wp.media to use the WordPress Media Manager dialog.

First, you need to enqueue the scritps:

// As you are dealing with plugin settings,
// I assume you are in admin side
add_action( 'admin_enqueue_scripts', 'load_wp_media_files' );
function load_wp_media_files( $page ) {
  // change to the $page where you want to enqueue the script
  if( $page == 'options-general.php' ) {
    // Enqueue WordPress media scripts
    wp_enqueue_media();
    // Enqueue custom script that will interact with wp.media
    wp_enqueue_script( 'myprefix_script', plugins_url( '/js/myscript.js' , __FILE__ ), array('jquery'), '0.1' );
  }
}

Your HTML could be something like this (note my code use attachment ID in the plugin setting instead of image url as you did in your answer, I think it is much better. For example, using ID allows you to get different images sizes when you need them):

$image_id = get_option( 'myprefix_image_id' );
if( intval( $image_id ) > 0 ) {
    // Change with the image size you want to use
    $image = wp_get_attachment_image( $image_id, 'medium', false, array( 'id' => 'myprefix-preview-image' ) );
} else {
    // Some default image
    $image="<img id="myprefix-preview-image" src="https://some.default.image.jpg" />";
}

  echo $image; ?>
 <input type="hidden" name="myprefix_image_id" id="myprefix_image_id" value="<?php echo esc_attr( $image_id ); ?>" class="regular-text" />
 <input type="button" class="button-primary" value="<?php esc_attr_e( 'Select a image', 'mytextdomain' ); ?>" id="myprefix_media_manager"/>

myscript.js

jQuery(document).ready( function($) {

      jQuery('input#myprefix_media_manager').click(function(e) {

             e.preventDefault();
             var image_frame;
             if(image_frame){
                 image_frame.open();
             }
             // Define image_frame as wp.media object
             image_frame = wp.media({
                           title: 'Select Media',
                           multiple : false,
                           library : {
                                type : 'image',
                            }
                       });

                       image_frame.on('close',function() {
                          // On close, get selections and save to the hidden input
                          // plus other AJAX stuff to refresh the image preview
                          var selection =  image_frame.state().get('selection');
                          var gallery_ids = new Array();
                          var my_index = 0;
                          selection.each(function(attachment) {
                             gallery_ids[my_index] = attachment['id'];
                             my_index++;
                          });
                          var ids = gallery_ids.join(",");
                          jQuery('input#myprefix_image_id').val(ids);
                          Refresh_Image(ids);
                       });

                      image_frame.on('open',function() {
                        // On open, get the id from the hidden input
                        // and select the appropiate images in the media manager
                        var selection =  image_frame.state().get('selection');
                        var ids = jQuery('input#myprefix_image_id').val().split(',');
                        ids.forEach(function(id) {
                          var attachment = wp.media.attachment(id);
                          attachment.fetch();
                          selection.add( attachment ? [ attachment ] : [] );
                        });

                      });

                    image_frame.open();
     });

});

// Ajax request to refresh the image preview
function Refresh_Image(the_id){
        var data = {
            action: 'myprefix_get_image',
            id: the_id
        };

        jQuery.get(ajaxurl, data, function(response) {

            if(response.success === true) {
                jQuery('#myprefix-preview-image').replaceWith( response.data.image );
            }
        });
}

And the Ajax action to refresh the image preview:

// Ajax action to refresh the user image
add_action( 'wp_ajax_myprefix_get_image', 'myprefix_get_image'   );
function myprefix_get_image() {
    if(isset($_GET['id']) ){
        $image = wp_get_attachment_image( filter_input( INPUT_GET, 'id', FILTER_VALIDATE_INT ), 'medium', false, array( 'id' => 'myprefix-preview-image' ) );
        $data = array(
            'image'    => $image,
        );
        wp_send_json_success( $data );
    } else {
        wp_send_json_error();
    }
}

PD: it is a quick sample written here based on other answer. Not tested because you didn’t provide enough information about the exact context the code will be used or the exact problems you have.