You can tell this by looking at the final line or two lines of the story. They do not literally come out and say to us "and then Rainsford killed Zaroff," but they do tell us what we need to know.
In the next to the last paragraph, the two of them square off to fight. Zaroff lets us know that the fight will be a fight to the death. He says that one of them is going to end up being fed to the dogs. But the other one will get to sleep in a very nice bed. So that tells us what's going on -- one dies, one sleeps in the bed.
Then the last line tells us that Rainsford thought that the bed was very comfortable. If he can think that, he is clearly the one who is alive and sleeping in the bed, not the one who is dead and being fed to the dogs.
Here are the relevant lines:
The general made one of his deepest bows. "I see," he said. "Splendid! One of us is to furnish a repast for the hounds. The other will sleep in this very excellent bed. On guard, Rainsford." . . .
He had never slept in a better bed, Rainsford decided.