On this 3x3 chess board, the black bishop can only ever move on a diagonal, which means it can only travel onto white squares: it can never land on a black square. Since the central square is white, the bishop has put the food in the middle.
The white knight can only make knight moves - from its starting point, it can only move edge-corner, edge-corner, edge-corner, edge-corner before coming back to its starting point. The knight can never land in the middle square at all, so it will eventually starve.
The knight can, however, move onto the black squares: in fact, it has to land on a black square every other move.
The bishop's claim that he was "that close" to capturing the knight is also exaggerated - it can be shown how, barring an error, the pieces could move about that board forever in an endless loop.
It appears to be a bunch of mixed gaming metaphors. "He stepped up to the plate and scored a touchdown.. checkmate" kind of thing.
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.