Passing variable from template file to the plugin file

You can use this way:

private function display_images( $args ) 
{
    $instagram_url = get_user_meta($user_data->ID, 'instagram_url', true);
    $username = $instagram_url;

 //You can use $username in another function by simply call it using global method.
 }

This is an easy way to get the post meta data in your custom plugin file.