Why does Gulliver cooperate with the Lilliputians?

As stated in the other answer, Gulliver is practical enough to cooperate with the Lilliputians for his own survival. But there are several other reasons for his willing cooperation.
First, the Lilliputians are tiny creatures, standing at only six inches tall. They seem delicate and innocent to Gulliver, like dainty little dolls. Like many people, he mistakes physical beauty for moral beauty or virtue. It takes him awhile to realize that although they look good on the outside, the Lilliputians are not good inside. Instead, they are petty, vain, and violent. Swift is satirizing the human tendency to confuse appearance with reality and the undeserved favor people who are beautiful receive.
Second, as we will learn throughout, Gulliver is, as his name implies, gullible, and he tends to take what he hears as truth. It is his nature to make quick judgments and accept what people say. In this way, he is like most people in the world. He also tends to accept what others say ahead of the reality of what he is experiencing, which is where much of the humor of the work come in: he is constantly praising things that the reader sees as ridiculous. Therefore, being who he is, it is easy for Gulliver to accept the Lilliputians at first at their own high valuation, and he wants to cooperate with them.
Finally, as with most people, Gulliver's best trait is also his flaw. We like him for his generosity of spirt and tendency to see good, but this also leads to his great flaw, his gullibility. Gulliver cooperates with the Lilliputians just because he is the kind of person to do so. Further, when he sees a problem, like a fire, he acts in a helpful and logical way because that is who he is. Like most people, he does not expect to find his good deeds punished, although, as happens in real life, they are.

Gulliver cooperates because he really does not have much choice in the matter. He has been tied down on the seashore by the Lilliputians. They may only be six inches high, but they are dangerous little critters; those arrows they fire really hurt. Gulliver has only just woken up, and as he is still feeling quite groggy and confused, he figures that cooperation is the best course of action. Gulliver also works out that the Lilliputians will treat him decently so long as he does not try anything. He indicates that he is hungry and thirsty, and the Lilliputians respond by fetching him baskets of meat and barrels of wine. Cooperation clearly pays off with these people. Now that his hunger and thirst have been sated, he wants the Lilliputians to go one better and set him free. Cooperation has already worked once, so it can work again.