For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
We meet the book's narrator, an unnamed captain who's only been in charge of his own ship for two weeks. His crew is a tight-knit group and he has a difficult time getting accepted into their fold. He also worries that they question his judgment because he's new and inexperienced. Anywho, the captain looks over the side of his ship one night and finds a naked man holding onto one of his boat's rope ladders. He finds out that the guy is a fugitive murderer, but lets him aboard anyway and hides him in his (the captain's) private room. The more they talk, the more sympathy the narrator has for the guy, whose name is Leggatt.
Act II
The narrator and Leggatt become close as the days go by. They whisper to one another about all sorts of things, although the narrator isn't always up-front about what these things are. Things get anxious, though, when the skipper and crew from Leggatt's ship come looking for him. The narrator is clever enough to throw them off the trail, and these dudes eventually leave believing that Leggatt drowned in the ocean after escaping their ship. The coast is clear for now, but Leggatt decides that sooner or later, he'll need to leave the narrator's ship. The best plan is for him to swim to one of the nearby islands, since there's no way he can return to Britain with the narrator. There'd be way too many people who might recognize him.
Act III
The captain-narrator goes on deck and orders his crew to steer their ship toward some nearby islands in the middle of the night. He plans on giving Leggatt the best chance of swimming for them, but he's also putting his ship and crew in grave danger by sailing so close to land with no light. His crew thinks he's gone bonkers, but they obey anyway. It's a super close call, but the narrator is able to pull the boat away at the last possible second. Once the ship is in the clear, he looks over the back of the boat and sees his hat (which he gave to Leggatt) floating in the ocean. He closes the story with the hopeful thought that Leggatt has made it to shore and will live a free and happy life.