What forces Jim to grow up and mature very rapidly is the takeover of the ship that is launched by Long John Silver. In Chapter 10 it is detailed how Jim hides in a barrel, only to find himself being able to over hear Silver as he discusses his plans. Note how Jim refers to this situation:
It was Silver's voice, and before I had heard a dozen words, I would not have shown myself for all the world, but lay there, trembling and listening, in the extreme of fear and curiosity, for from these dozen words I understood that the lives of all the honest men aboard depended upon me alone.
It is the responsibility that Jim suddenly has that catapults him from being a mere cabin boy, the lowliest member of the crew and a child, to being an adult who holds the lives of others in his hands based on what he does and does not do. Jim suddenly finds that he has to mature very quickly, as he has to cope with a very dangerous and risky situation. It is this moment of understanding and this sense of responsibility that results in his coming-of-age, and it is clear from the novel that Jim, after these events, is completely different from the youthful Jim before these events. Jim, at the end of the story, clearly is presented as an adult, who has proven himself as a man and shown his maturity. It is the responsibility that Jim has thrust upon him therefore that triggers his rapid maturity, as he has to face the fact that his actions will determine whether others live or die.
He made a lot of choices on the board, for example he could've easily joined the pirates and leave the doctor behind. He also matured through fighting. Since he fought with the pirates, he probably gained a lot of muscles. From one scene where they fought on the ship and almost got his head chopped off by a knife. He probably gained some reflex skills and acrobatic skills as well. Also lastly of course he aged, since they were on a ship for a long time!