Saying "Hold on tight!" is what anyone might do when carrying a child piggyback. Jesus loved and enjoyed children, and probably played with them from time to time. It is not hard to imagine a group of children trying to get His attention and one feeling very special riding high on His back.
The direction to "Hold on tight!" may also be taken metaphorically – do we follow His teachings with all of our might? His teachings and commandments were given to help us avoid problems and ultimately be happy. The message is as simple as the painting; Christ wants us to be happy simply because he loves us.