Rainsford learns two important lessons in the short story.
First, Rainsford learns what it means to fear. This is an important lesson, because in the beginning of the story Rainsford is very caviler about fear. When Whitney says that he believes that jaguars fear, Rainsford dismisses him. Rainsford says: "Nonsense." Afterwards, he laughs at Whitney's opinion.
When Rainsford is being hunted on the island by the crazy General Zaroff, he learns fear. He knows what it means to be a hunted animal. Here is what the text says:
"First, At daybreak Rainsford, lying near the swamp, was awakened by a sound that made him know that he had new things to learn about fear."
The text makes this point even more explicit, when it says:
Rainsford knew now how an animal at bay feels.
The second point that Rainsford learns is that some people, like Zaroff, are crazy. Rainsford is utterly horrified at Zaroff's new "game" of hunting humans.
Most interestingly, it is revealed at the end of the story that Rainsford kills Zaroff. Perhaps Rainsford also learns that he can kill a man and enjoy his sleep. Here is how the story ends, with Rainsford the only one remaining:
"The other will sleep in this very excellent bed. On guard, Rainsford." . . .
He had never slept in a better bed, Rainsford decided.