Monstro is the giant whale from the 1940 Disney movie Pinocchio. When you're done with Agrabah, get on the path to Atlantica, where a cutscene will show Monstro intercepting the way to your original destination. Before the party or the Gummi Ship is able to react, the whale will swallow the ship whole, thus trapping you inside its body. Expect a maze when you begin to enter the different chambers of his body. This level plays similar to Jabu-Jabu's Belly from The Legend of Zelda: Ocarina of Time.
A cutscene will show Pinocchio, who you saw back at Traverse Town. Jiminy Cricket, along with the party's curiosity, will lead you to follow Pinocchio. Swim to the platform on the left, in front of a big pile of discarded pieces of wood. After another cutscene, you'll be reunited with Pinocchio and find out what he's up to. You'll also be introduced to his father, Geppetto. While Geppetto is discussing matters with the party, Pinocchio wanders off into a tunnel that leads to Chamber 1. Follow Pinocchio into the chamber.
The Chambers seem to be the bulk of Monstro and one of the most confusing parts of the game. Getting through them requires a lot of trial and error, along with memorization. You will now enter a part of Monstro's body covered in what seems to be colorful acids. After a cutscene you'll find out who's behind Pinocchio's troublesome behavior - Riku. He'll now make you find him, both for answers and Pinocchio. From here, take the left exit to Chamber 2. Quite simply, here you must go through the exit leading to Chamber 3. You'll end up in the upper-level of Chamber 3. The Heartless here will try to get you off the ledges by flying away from the ledges. Pursuing them will lead to having to start over again. Take the path to Chamber 2. Go to the exit to Chamber 5. Go to the exit to Chamber 6. Go down the steps here and then east to Chamber 5. Jump on the barrel and then to the ledge. This leads to Chamber 4, which will be followed by a cutscene.
After the battle, you'll be back at Geppetto's ship, where he'll give you the High Jump ability. With your new ability, you'll be able to reach an entrance in front of the entrance to Chamber 1. This leads to the Throat, where your second and final boss battle for the world will be.
Immediately after the boss is over, because of the disruption you're causing in Monstro's body, he'll spit you out and you'll end up back on the Gummi Ship.