Why did the Puritans come to America?

The Puritans came to America primarily because they wanted to practice their religion in peace. At that time in England the Puritans were persecuted because they wanted to purify—which is where the name Puritan comes from—the state Church of what they saw as Catholic practices.
The political and ecclesiastical hierarchy regarded the Puritans as dangerous trouble-makers who represented a threat to the stability and security of the state. In those days, there was no real separation between politics and religion, so if a particular religious group such as the Puritans had different ideas of how the Church should be run they were, at the same time, also putting forward an alternative political arrangement, in clear opposition to the status quo.
Once Puritans realized they could no longer reform the Church of England they sought to found their own churches. This was only possible where the state authorities were Calvinists like themselves—such as in the Netherlands—or where there was a separation of church and state, as in America.