The Pilgrims were separatists from the established Church of England. This placed them on the radical fringes of groups arguing for religious reform in England in the late sixteenth and early seventeenth centuries. Some reformers, known as Puritans, wished to purge the Church of what they thought were corrupt elements, but the Separatists thought, in short, that the Church was too far gone, and that the only appropriate course of action was to leave it.
Facing harassment from the Church (remember that there was no such thing as separation of church and state in England at the time) they left for Leiden, in the Netherlands seeking a safe haven. But they found this solution unsatisfactory for a number of reasons and eventually decided to establish a permanent colony in North America, which had already been settled by the Virginia Company at Jamestown. The idea, of course, was to leave the church and gain religious freedom.
To raise capital for the endeavor, they formed a group of investors and purchased ships and supplies. When they arrived, they settled in an area north of their intended site, and it was at this point, in modern Massachusetts, that they established Plymouth.