There are at least two reasons why the Pilgrims felt that they needed the Mayflower Compact.
First, the idea of covenants was very important to them. Puritans saw their relationship with God as one that rested on a contract or covenant. Therefore, it makes sense that they would want to have a covenant to bind their community together.
Second, and more practically, they felt the Compact was needed to establish a legal basis for their government. They were not going to settle in the place (Northern Virginia) that their charter called for. This breach of their charter led some to argue that the whole government of the colony was invalid. Therefore, they felt they needed this Compact to give the government a legal foundation.