Why did the Pilgrims come to America?

The Pilgrims came to America primarily because they wanted to practice their religion in peace. In contemporary England, there was widespread persecution of those Protestants—generally known as Puritans—who wanted to purify the established church of what they saw as remnants of Catholicism.
Unfortunately for them, the authorities of both church and state were profoundly hostile to such moves, seeing them as undermining the established social order. Most Puritans remained members of the Church of England, trying to change it from within, but a minority felt that they had no future left in England and set out to establish a godly kingdom in America, where they were sure that they could establish a church of their own, free from the control of the state.