I know that this is fodder for all kinds of conspiracy-theory BS, and to be crystal clear, I want none of that here. I don't believe, nor would I particularly buy, into claims about some global cabal of Masons controlling the strings of government. But, some of the founding fathers were most certainly Masons, and as I understand, Masonic Lodges definitely did have connections with Federalist political groups in the period, so I'm interested in what the relationship was there, and how Masonry was actually related to American politics at the time.
Freemasonry, as an organization, was less involved than you might think.
There were certainly many influential historical figures who were also Freemasons - I haven't been able to walk ten feet into an American Lodge building without encountering a portrait of George Washington. There has been debate, both within and without the Fraternity, as to whether these great men joined Freemasonry because it echoed virtues that they already held, or whether Freemasonry encouraged those virtues in them once they joined. Likely a bit of both, but that's not really relevant.
One of the earliest guiding documents of Freemasonry was The Constitutions of the Free-Masons, compiled by Dr. James Anderson and first published in 1723. It includes the following admonition:
Therefore no private Piques or Quarrels must be brought within the Door of the Lodge, far less any Quarrels about Religion, or Nations, or State Policy, we being only, as Masons, of the Catholick Religion above mention'd, we are also of all Nations, Tongues, Kindreds, and Languages, and are resolv’d against all Politics, as what never yet conduct'd to the Welfare of the Lodge, nor ever will.
(It is important to note that the use of the word “Catholick” in this instance refers to its traditional definition of “inclusive” and is not a reference to Roman Catholicism.)
There are two reasons for this prohibition, for which Anderson's Constitutions gives a little background. The first is that a Masonic Lodge is supposed to be a place of harmony between its members. Freemasonry entered public perception in the aftermath of great conflicts in England - between Catholics and Protestants, between Parliamentarians and Royalists, etc. - and contemporary Freemasons wanted the Fraternity to be a safe space where these contentious opinions and issues could be left at the door and its members could focus instead on what they had in common.
The second reason is that the forefathers of the Fraternity recognized and appreciated the terrible danger of organized sociopolitical action. Freemasonry quickly spread around the world as merchants, soldiers, and dignitaries rounded the globe, resulting in a large network of loosely-affiliated Masonic jurisdictions. They were united by a common history and shared ancestors, but found themselves in vastly different social and geopolitical environments. The activities of one jurisdiction could (and, indeed, did) color the public perception of the Fraternity as a whole and a bad reputation developed in one area of the world could later harm members in another. It was wisely decided to leave matters of activism to the professionals.
So, while there were a number of high-profile Freemasons involved in the American Revolution, the Masonic Lodges themselves stayed out of it.
Unfortunately, some branches of the Masonic family tree (it is not a cohesive organization) did become directly active in social and political movements. The participation of the Grand Lodges of Germany in the Kulturkampf of the 19th century is one example, but this contrast might be going too far beyond your original question.
Further reading: