Explain why “the garden of forking paths” can be seen as a “three-dimensional chess game.”

This is because Yu Tsun is more melancholy and full of foreboding. He says lots of ominous, uncheerful things, like "with the eyes of a man already dead, I contemplated the fluctuations of the day which would probably be my last." Since he's the narrator we hear from the most, the overall tone is one of gloom and doom. The story shows that there are many challenges as the story develops, but in the end he wraps up every little loose end to his labyrinthine narrative in the space of a single paragraph – and in this case, there's a lot to wrap up. 

Your question is the answer