As Rainsford finds himself in a game of life and death, he has no option other than to play the game to the best of his abilities. At first, he is mainly defensive, but as he is being hunted, he realizes that he needs to be on the hunt as well. Perhaps, he realizes what he said in the beginning of the story that there are only two classes in the world, the hunter and the huntee.
Whatever the reason is, Rainsford begins to go on the hunt. He realizes that waiting or fleeing are only delaying the inevitable. So, he decides on an alternate strategy, which leads him to swim to Zaroff and wait for him at home. In other words, he is on the offensive. Here is what Rainsford thinks to himself:
Rainsford knew he could do one of two things. He could stay where he was and wait. That was suicide. He could flee. That was postponing the inevitable. For a moment he stood there, thinking. An idea that held a wild chance came to him, and, tightening his belt, he headed away from the swamp.
In light of these point, Rainsford's survival was made possible by his decision to alter his plans and embrace the game by hunting.