I assume you mean the Pilgrims, not Puritans; they were two different groups of people to start with. The Pilgrims came to the New World because they were not allowed to hold public office in Great Britain, and a group of them through political and economic infuence managed to get a sort of dispensation to come to "New England" and create their own colony in which they would be allowed to hold offices and do all the things to run their own society which they were forbidden to do in England. They did not come to America to create a colony in which everyone had the freedoms they wanted, they came to create a society for themselves. Their attitude was that other groups could try to do the same thing which they had done. For instance, they were not tolerant of Quakers, but Quakers eventually founded their own (much more tolerant) colony in Pennsylvania.
The colony of Massachusetts was the first British colony on the North American mainland to legalise chattel slavery (1644), officially believed that Native Americans were not human and had no souls, and were the starting point of the racial theory that Africans were inferior to Europeans, and thus slavery was justifiable. They were not always tolerant people, but tolerance is not a trait that was notable in any European society at the time. It's not really surprising, therefore, that they acted as they did.
First, they left England to avoid such a society, but that doesn't mean they were opposed to intolerant authoritarian societies. What they really wanted was an intolerant authoritarian society that THEY controlled.
Second, the reason they felt that way was religious. They believed that God would punish (on Earth, not just in the afterlife) any society that allowed people to behave in incorrect ways. So they wanted everyone in their society to act correctly so God would not punish their society for allowing sin. This is why they did not want to tolerate any dissent or "wrong" action.