The colonists came to America: to build a better economic life for themselves and to have religious freedom. I will focus on the second reason and how it played out in the decision of the Mayflower pilgrims to migrate from England to found the Plymouth Colony in Massachusetts.
James I was the monarch in Great Britain during the period of the Puritan migration to Plymouth. He very much wanted to impose religious orthodoxy in order to legitimize his rule and insisted on adherence to Church of England standards. He tightened up and bore down on dissenting religious groups in a way Elizabeth, on the whole, had not.
As a result of increased persecution that included jailing, fines, and land seizures, the Puritans decided to migrate to Holland, which offered religious freedom. This they did, but realized over time that their children were assimilating to Dutch culture. As they wanted very much to keep to English distinctiveness, they returned to home. Facing intense persecution in England, they decided to migrate to North America. They desired religious freedom. This was not religious freedom as a general principle to be offered to any group but freedom to establish their own form of rigid theocracy. They assumed that in the New World they would be able to obtain both freedom from religious persecution and be able keep their distinctive English culture. In both cases, they were correct.
It is important to keep in mind that the Puritans found remaining an English cultural identity a very high priority. They toiled in 90 degree temperatures in woolen clothes and built English-style house when Native American buildings would have been more practical—not because they didn't realize the problems but because they actively wanted to remain British.