The Puritans were a minority sect of Protestants who wanted to purify and separate from the Church of England. They were part of a larger group of Christian sects in England (called Dissenters) who thought the official English Church was corrupt and had moved too far away from Biblical precepts.
The majority of the English, who were members of the Church of England (also known as the Anglican Church in the United States), viewed the Puritans with suspicion as radicals who wanted to upend British society and government. They were perceived much as communists might have been in the United States during the Red Scare—a group outside of the mainstream whose intentions toward the government and their fellow countrymen were suspect.
In Great Britain, church and state were intertwined. The king, the head of state, derived his legitimacy from the Church: he was understood by English society as specially appointed and anointed by God to rule. The Church of England, with its bishops and hierarchy, upheld his divine right to reign. As James I said, "no bishops, no king."
The Puritans didn't accept the authority of the Church of England or its bishops, so they were suspected of not supporting the king. They were seen as potential traitors and troublemakers whose motives were unknown. Many British were glad to say good riddance to them and ship them off across the ocean, out of their hair.