In the game, a child is told to go to sleep and is pulled out of bed and through the window, into a world of dreams. In this dream world, Doshin is now imprisoned and you can "Tinkle" 2D hearts on people and even the giant himself. The main objective of the game is to watch the 17 mini black and white movies collectively titled "More Than Giant" that were specially animated for this game. You must repeatedly go back and forth between the two disks to verify that monuments were built in game 1, and to complete tasks that the "Queen Companions" request of you. Most of the mini-movies in this game are simply stand-alone animated shorts called "more than giant' telling bits of the story of how Doshin came to be imprisoned (e.g. a boy fishing catches a giant, Doshin raises and lowers land to the sound of rap music, etc.).
Another objective is to free Doshin from his imprisonment, by causing him to grow larger than his cage. This will cause the game credits to show and can only be viewed once, unless you reset the game's saved data.
You can also gather help to free Doshin. After creating monuments in 'Doshin 1' and their counterpart pavilions, children will appear in the Expo area and you can "tinkle" on them with your hearts and they will join you, becoming what are known as Teamers or teamsters. There are two other teams you can choose from at the start of the game, and with enough pavilions completed you can battle the other teamster groups in a tinkle contest. If you win, the loser's team members become neutral and can be added to your team. You can have 6 other children on your team. This gives you 6 times more tinkling hearts when you go to try to free Doshin.
Source: Wikipedia, "Doshin the Giant", available under the CC-BY-SA License.