Clayton has finished his Skybax training when his friends Hyla, a human female, and Krekor, a Hadrosaur, inform him of a threat to Dinotopia. A band of pirates are stealing and hiding Tyrannosaurus eggs all over the island, attempting to draw the creatures from their nesting grounds so they can search for an artifact of some sort. Clayton is given a Sunstone Prod, a device similar to a staff, to use against the pirates as he hunts for the eggs.
As the game progresses, it becomes apparent that the pirates are after the Timestone, a mythical artifact said to slow time around whoever holds it. Clayton must race to find all of the remaining eggs and locate the Timestone before the pirates do.
Each level takes place in a different part of Dinotopia. After three sections of each level, Clayton faces an extra challenge, usually a boss to fight, and then must return the eggs he recovered to the Tyrannosaur nesting area without getting killed before proceeding to the next level.
Source: Wikipedia, "[en.wikipedia.org/wiki/Dinotopia%3A_The_Timestone_Pirates Dinotopia: The Timestone Pirates]", available under the CC-BY-SA License.