Probably the principal theme of the story is the connection between all living beings, regardless of their intelligence or power. Rainsford begins by judging the animals he hunts as lacking attributes that we conventionally identify as "human." He does not even believe an animal can feel fear. The turning point in the story comes when Rainsford, as he is hunted by Zaroff, reflects that he now "knows how an animal at bay feels."
A second theme is the obvious one that "whatever goes around, comes around." Rainsford is arrogant and unfeeling, having made his career of hunting for sport, never thinking that the same aggression could someday be inflicted upon him. The fact that he wins the battle, however, means that in some sense the lesson is invalidated. The outcome of the story merely proves that the strong survive, which is what Rainsford believed all along. It comes down to a match of power versus power, man/beast individually against man/beast. Whether the author intended this message, we can't really know. Rainsford does seem to learn empathy, but only within the context of the mindset that has governed him from the beginning.
A third theme, though a more ambiguous one, involves the human transference of aggression from one object to another. Zaroff is a Russian émigré , having fled his country after the débâcle (the Russian Revolution), as he terms it, but having landed on his feet because of his investments in "American securities." So he has built this palatial home on an island in order to kill people. His attitude is one of displaced anger, directed against the helpless. The fact of his having constructed this based upon "American" investments is a subtle commentary on a general human complicity in violence, and creates a parallel between the dog-eat-dog world of economics and the hunter-animal world of the jungle.