How did the Puritans die out?
Puritan philosophy never really died out; one could argue that institutions founded during their time, such as Harvard and Yale, continue to promote some elements of their philosophy established in the early 1600's. In terms of religious belief, the Congregationalists are the direct descendants of Puritan theology. In terms of politics, as the ruling group in colonial New England, Puritan authority remained supreme until about 1700, when the Salem Witch Trials destroyed the theocracy, which, along with increased immigration of non-Puritans who did not adhere to their religious beliefs, weakened Puritan influences. The governmental system established after the Witch Trials was more democratic; and became more so up to and beyond the American Revolution. However, even early on, Puritans relied on town meetings as a form of direct democracy; their establishment of even a limited democracy which became stronger and widespread as the country expanded is perhaps their most important gift to us.
Here is another perspective. Puritan theology and way of life did not die out at all. One can make a strong case that many conservative presbyterians, baptists and congregationalists are all direct descendants of Puritan theology and way of life. In America, there are also many thriving seminaries that hold onto Calvinistic teachings (a very Puritan thing to do), such as predestination. Just look at Westminster theological seminary in Philadelphia and California, Reformed theological seminary, Trinity Evangelical Seminary, and many more. All of these schools find a lot in common with Puritans. So, the question is a bit odd. It assumes death, when there is great continuity even now. Finally, as an example, I added a link of a book review of someone in the Calvinist camp.