This is all according to Leon Botstein.

The piece was occasionally played for July 4th celebrations, simply because it was as Botstein says, "a barnburner," and one of the few pieces which calls for cannons. That makes it uniquely easy to adapt for any military celebration. So were were already periodic performances throughout the country for the 4th.

The key moment though, was when in 1974 the Boston Pops did a large extravaganza 1812 performance. It included not only the traditional cannons, but it was accompanied by a fireworks show, and a bell choir was created using local church bells. It was a grand performance by all accounts... and it was nationally televised. Over the next few years it was picked up and performed by orchestras across the country in outdoor performances.

I was already halfway done with this response when I saw you got another answer, but I'll put up what I had anyway. Tchaikovsky originally wrote it for a number of celebrations happening in Russia in the early 1880s including the Moscow Exhibition, and in reality, it is a fairly straightforward example of program music that follows the 1812 campaign to its victorious end. A lot of people in the United States don’t realize that it was actually focused on Napoleon’s demise to celebrate the 70th anniversary of Russia’s victory. The piece features motifs of La Marseillaise, Russian folk music, and the Tsarist national anthem. According to music historian Roland John Wiley, it also “warrants attention for quoting a liturgical melody (the opening theme) and the national anthems of Russia and France. The brassy statements of ‘La Marseillaise’ are another mask for personal memory, echoing Litolff’s Ouvertüre zu Maximilian Robespierre, a favourite work of Tchaikovsky when he was a student.” Tchaikovsky himself described the piece as “very loud and noisy” but didn’t particularly like it or find it to be his best work.

The piece was played now and then in the United States, including its 1891 performance at Carnegie Hall directed by Tchaikovsky himself and in 1927 in Cleveland where it was performed indoors with percussion rather than cannon. It also had notable performances internationally in Britain, Germany, and the Soviet Union. As /u/QuickSpore wrote in his or her response, it really became a staple of 4th of July celebrations when the Boston Pops Orchestra under the direction of Arthur Fiedler performed the piece and broadcast it in 1974. The piece captured the audience’s attention as the nation came into its bicentennial, and it’s been a classic 4th of July piece ever since.

Interestingly, the 1812 Overture went from being a quintessentially Russian piece performed frequently at “Tchaikovsky Concerts” to a classic example of Americana right in the middle of the Cold War. I think this transition has more to do with programming than history though. The piece is loud, fast, and thrilling. It is fantastic to perform in front of large crowds at an outdoor celebration because it is an absolute spectacle. Brilliant brass fanfares, fireworks, cannon, chimes, and bells--the piece has an air of victory, celebration, and triumph. Furthermore, most ensembles only play the climax of the piece and leave out much of the build-up, which takes a lot of the contradictory historical aspects out. It also has a recognizable melody at the end, which draws the audience to the piece. In this way, the piece became more malleable for other purposes like 4th of July celebrations. However, keep in mind that the piece was and is performed all over the world as a celebratory or just plain fun piece for both indoor and outdoor spectaculars.