in the beginning of the story Rainsford was on a yacht in the sea with his crew going to go hunt leopards.
In the hook of the story it was a foggy night and Rainsford was getting some sleep he fell off the yacht and was floating in the middle of the sea as if he had no hope of them ever coming back for him and he was lost in sea and had no choice but to swim to the island he was warned not to ever go to or stay out there with the creatures of the sea
He kept hearing gun shoots and that was making him curious rather to go that way and check out what was going on rather than be stranded in the sea without anyone noticing he was gone. Also he met Zaroff whn he made it to the island and Ivan.
Zaroff was basically getting Rainsford prepared to be hunted down not at this part. So then that was when he gave Rainsford some hunting stuff and sent him into the night to hunt him down.
While Zaroff and Ivan were hunting down Rainsford with the dogs on his scent Rainsford had a trap set up that took Ivan's life. Then when Rainsford was on the move the dogs were on his scent so he had no choice but to jump off the cliff or he would die.
While Zaroff was in disappointment of losing his target and not figuring out how to replace Ivan Rainsford managed to get in Zaroffs home before he did. And Zaroff said " one shall sleep a good night and one shall be fed to the dogs" so Rainsford basically kilt Zaroff at the end.