The objective of the minigame is to defeat Heartless until the meter on the screen is completely filled. When the meter is filled and all of the Heartless are destroyed, the minigame is over. Monstro then sneezes Sora and his friends out of his body, and the world is completed. The minigame has to be completed in order to continue with the game.
While this is considered a mini-game, there is no secret to beating Monstro's Belly Brawl. The thing that should be noted is that the meter increases with each Heartless defeated, and decreases with every second the player doesn't defeat a Heartless or stays idle, Meaning that the player has to defeat the Shadows consecutively, until the meter fills completely.
Stay clear from Sleights or Summons if you want a consecutive attack going round so the meter doesn't drop.