Show message from backend

The problem is that when a post is saved, there’s a redirect back to the edit post page and the admin_notices action never runs. This means that any admin notice you put on any of the save post actions (save_post, transition_post_status, etc.) will be thrown away (since they’re not persistent) during that redirect.

The workaround is to save a state (user settings/user meta, post meta, transients, options) which would signal that an admin notice should be shown, and look for that state on the next request.

add_action( 'save_post', function( $post ) {
    // some checks here
    update_user_option( get_current_user_id(), '_my_show_notice', true );
});

add_action( 'admin_notices', function() {
    if ( get_user_option( '_my_show_notice' ) ) {
        delete_user_option( get_current_user_id(), '_my_show_notice' );
        echo '<div class="notice notice-success"><p>Some notice</p></div>';
    }
});

Hope that helps!