That's the "backstory", the main joke is just about how the chess pieces move, i.e. that the bishop can't ever get into a black square and that the knight can't

That's the "backstory", the main joke is just about how the chess pieces move, i.e. that the bishop can't ever get into a black square and that the knight can't ever get to the food. I think the rest of the dialog is just the scenery of a world where game pieces talk.