Freemason and amateur history buff when it comes to fraternalism in the US, here.

Freemasonry has had a complicated relationship with US politics. I'll touch on a few high (and low) points just to introduce the topic, but you can follow the rabbit hole quite a lot deeper if you choose (even just Wikipedia surfing can be quite useful).

Late 18th century: The Grand Lodge of England (now called the United Grand Lodge of England) was formed in 1717, bringing individual Lodges together into an organizational structure for the first time (note: this is a claim, but it's argued by, e.g., Scotland). In the 1730s, they authorized "provincial" Grand Lodges in the US, which is to say satellite administrative bodies that were still seen as part of the Grand Lodge of England. However, over time these Grand Lodges sought more and more autonomy, and eventually each colony would have its own Grand Lodge, acting with sovereignty over its Masonic jurisdiction.

Freemasonry in the US was fairly popular, and the meetings, at least in Massachusetts and Pennsylvania, appear to have been a nucleation site for the some of the relationships that would become the bedrock of the US revolutionary war effort. It's important to remember that many Freemasons were loyal to the British, and many thought that Masons like Franklin and Washington were in the wrong. Many did not. Many founding fathers were not Freemasons. Real history is complicated.

After the Revolutionary war, however, this association stuck in the public's mind, and Freemasonry became a serious political point of reference. There were many places in the early US where potential politicians joined the Fraternity just for the tacit association with the founders.

Early 19th century: Freemasonry continued to be a prominent symbol in politics (though it's not clear that there were any Lodges or Grand Lodges interested in actually playing politics). This was a state of affairs that some were not pleased with, John Quincy Adams being one of the more visible examples, and when, in the 1920s, Freemasons in upstate NY were embroiled in a controversy around the disappearance of someone who printed an expose of their rituals, he and the so-called "Adams men" were among those who sought to promote and capitalize on the scandal. The "Anti-Masonic Party" was founded on the back of this scandal and went on to be one of the most successful third parties in US history.

The so-called anti-Masonic period lasted until the late 1930s and saw many Lodges along the East Coast especially, disbanding. When Freemasonry resurged it would take it decades to reach its previous heights. When it did, it was on the back of a surge of fraternalism that saw hundreds of new organizations spring up. In some places, groups like the Odd Fellows were actually larger than Freemasonry. As the 19th century came to a close and the 20th began, fraternities like Freemasonry were seen as one of the ways that men integrated themselves into society and learned to be leaders.

The Great Depression was the definitive end of the so-called golden age of fraternalism, and though Freemasonry did not shrink as much as it had grown, its influence during that period was vastly reduced. However, the fact that the President of the United States (FDR) and the man who stepped in to replace him on his death (Truman) were both Freemasons and were tightly linked to the end of the Depression and the winning of World War II, lead to another public surge in joining the Fraternity. Membership was at an all-time high per capita and in raw numbers in the 1950s and 1960s, and so the influence of the Fraternity was again at an all-time high. After this period, membership in many public societies began to decline, and Freemasonry saw a steady (surprisingly so, actually) decline in members from 1960 until the early 2000s.