I would say that the role of the Mayflower Compact was that it A) set up the first democratic settlement in the New World and it B) allowed the society of Plymouth to be established on a stable basis.
Not all of the people who went to Plymouth on the Mayflower were Puritans. This meant that there could potentially be some conflict among the settlers as to how the colony would be set up and run. By creating the Mayflower Compact, the settlers avoided this. They set forth a miniature constitution that said how they would be governed. When the people signed it, they were all agreeing to abide by this system. That set up a consensual and stable basis for their society.