Much of New England was settled by Puritans, who were very socially conservative even by the standards of the time. How is it that New England (and Massachusetts in particular) slowly became one of the most progressive and socially liberal parts of the country? When did that shift start to occur and when would you say was the turning point?
Massachusetts has always had an anti-authoritarian streak. The Puritans were a movement against the state-sponsored Church of England. Despite the socially conservative nature of the Puritans in some ways in Christian practice, from the start, this didn't carry over into conservative/status quo politics. In fact, the opposite was true. In 17th Century England, the King and Royalists in Parliament didn't call Puritans "Puritans"--that was the name for themselves. They called them the "Dissenters" or "Separatists" (1, 2, 3, 4).
The Puritans/Dissenters in Parliament in the 1640s led the anti-Royal movement. In the lead up to the English Civil War, King Charles tried unsuccessfully to arrest five of his leading Parliamentary critics, all of them Puritans. This was after Puritan leader and MP John Pym proposed the Grand Remonstrance critical of the King, which was passed by Parliament in December 1641. (See: "The Five Members".)
Puritans continued to be the leading force against the King during the English Civil War. The New Model Army was made up of soldiers affiliated with or at least sympathetic to the Puritan cause. Their military commander who replaced the king, Oliver Cromwell, was likewise a Puritan.
The Puritans in Massachusetts were a little bit more cautious than their counterparts back in England in regards to the war, preferring to try to remain officially neutral during its early years so as not to get their supply chain cut off or suffer other repercussions if they supported the wrong side. Nonetheless, when push came to shove, the "bias of the people [of Massachusetts] was unfriendly to the King." When Cromwell's forces won, the Massachusetts government did not wait long to recognize his authority as legitimate.
By contrast, Virginia was said to be "the last of all the king's dominions that submitted to the usurpation [by Oliver Cromwell and the Parliamentarians] and afterwards the first that cast it off."
It should be noted that the makeup of the population as well as land ownership had a lot to do with it. Massachusetts was founded by Puritans for both religious and political reasons, so the people who came over in the early days tended to be small-time farmers, owning 40 acres of land or less. Virginia, meanwhile, had large tracts of thousands of acres that had been granted to various nobility, and so was much more Royalist. (That's also the beginning of the story as to why the South ended up with plantations and then slavery, while the North did not.)
Massachusetts had another motive for staying out of the English Civil War. They had established their own government by their own laws, under the authority of the king, and didn't want to risk angering either side and have their government forcibly dismantled.
The precursor to the Massachusetts state constitution was the 1641 "Body of Liberties" drafted by Puritan minister Nathanial Ward. Among other things, this document established under law some very liberal (for the time) rights for the Massachusetts citizenry, including the right to due process, a trial by jury, and no double jeopardy or cruel and unusual punishment. It also forbade the clergy from simultaneously holding public office, an early attempt at separating church from state. Many of these were radical ideas, rights not yet protected back in England.
A century and a half later, Massachusetts was still arguably the most liberal of the Thirteen Colonies. Many of the laws passed by Parliament back in Great Britain in the 1760s and 70s that led to the American Revolution were specifically passed in response to the actions of the anti-Parliamentary Massachusetts government and citizenry. Among them was the "Boston Port Act" which led to the Boston Tea Party in 1774.
The other New England states share a similar history. New Hampshire and Maine were carved out of the Massachusetts colony/state, while Connecticut and Rhode Island were founded by Massachusetts Puritans who had early divisions with the Massachusetts colonial leadership, though both were still firmly on the side of the "Dissenters".
TL;DR: There was no "transformation". The Puritans as a political force were known by their opponents in England as the "Dissenters" or "Separatists". They were one of the leading liberal political forces in England in the 1600s. This liberalism carried over to the establishment of New England starting in 1620, which was split into several colonies/states later on.