Here’s where we get the crucial set up: Rainsford is a big-game hunter who thinks he’s all that. The animals? Pfft. It's not like they have any feelings about being hunted. So he may not be the most likable guy—we definitely know what we're getting with our protagonist.
Once Rainsford falls in the water, he doesn’t have the safety of his whole “I’m a hardcore hunter smoking a pipe on a yacht” attitude any more. Now it’s all he can do to get to the safety of the shore--so why not swim in the direction of those pistol shots?
So Zaroff may serve foie gras and champagne, but he also wants to hunt down his guest like a beast. So we have a little reversal of fortunes here, as Rainsford now finds himself in the position of the prey.
Rainsford does his derndest to elude Zaroff. But that Zaroff is good. Rainsford uses all of his old hunter’s tricks and then finally just uses his wits: he jumps into the ocean.
Well, turns out Rainsford survived his leap into the sea—and he’s mad. Real mad. So he does what any good vengeful hunter does—especially one who doesn’t believe in, er, killing people—he kills Zaroff. Wait, wait—but he lets the dogs do the really dirty work.