There were a few attempts to restage and reimagine the Nutcracker in the early 20th century, but it took two American productions to firmly lock the Nutcracker into Christmas tradition -- and help make ballet into a serious art in America.
On Christmas Eve 1944, the San Francisco Ballet staged the first full American version of the ballet, and it was a sensation. Staged by William Christensen to appeal to American audiences, it proved an immense hit, and SFB retains the claim to having the first American Nutcracker, restaging it again and again every decade or so to keep it up to date and fresh for the dancers and audiences. The Christensen version, however, migrated to the company he founded, Ballet West, in the 1960s, and they continue to present his version.
However, George Balanchine's Nutcracker proved to be the American Nutcracker when he had New York City Ballet present his version, which harkened back to the grandeur and scope of the Imperial Ballet he remembered from his days in training in Russia. The premier in 1954 featured Maria Tallchief as the Sugar Plum Fairy, and Balanchine tailored the choreography for her (as he always did with his chosen ballerina-muse). It was an instant hit and became an annual repertoire tradition the next year. By the mid-1960s, most American ballet companies were staging versions of the Nutcracker every Christmas, and today, Nutcracker revenue accounts for anywhere between 40-60% of a company's revenue.
But why Nutcracker? There's no real good answer. The setting of Christmastime is an obvious contender. The cozy domesticity of Act I with the magical portal story to the Land of the Sweets and the ending of Clara waking up safe in her bed with her family is a story that's reassuring and ends in a way that emphasises the primacy of the family home and the family celebration. The abundance of children's parts makes it an easy vehicle for featuring ballet students, which is a draw for parents and family -- after paying for years of ballet class tuition, you want to see your kiddo graduate from a party guest to a tin soldier to maybe dancing a divertissement. It just hits a lot of notes that resonate with Christmas in the West. Obviously, it's not without its problems -- the Chinese/Tea and Arabian/Coffee divertissements are incredibly racist, and Balanchine admitted that his version of Arabian was basically a sultry belly dance sequence he put in there "for the fathers in the audience." [Dunning, Jennifer (26 November 2004). "Staying on Their Toes for 'The Nutcracker,' Show After Show". The New York Times.] There's been a lot of outcry about the racism in Nutcracker, and it seems inevitable that as ballet reckons with the conspicuous absence of Black dancers in principal positions and the traditions of blackface and other offensive materials, Nutcracker will also be re-evaluated to stop stereotyping Chinese and Arab people. Phil Chan's Final Bow for Yellowface is a good resource for reading on that.
Basically, Nutcracker had good bones, as do most Petipa ballets, and some savvy choreographers/company directors in America saw a chance to sell ballet to the masses with a fundamentally old-fashioned Christmas message in the ballet. The Tchaikovsky score being magnificent certainly helps, and in the mid 20th century, there was a cultural push to make Christmas a children's holiday, to create new traditions, to find ways of cementing Christmas in American culture as a consumer venue as well as a celebration of home, family, and children, all of which Nutcracker offers. It was a good fit for the moment, which Christensen and Balanchine clearly recognised. From there, other ballet companies around the world began to take notice. Nureyev choreographed a version for the Royal Ballet in 63, and the (then) Kirov and the Bolshoi renewed the Nutcracker and made it suit Soviet circumstances. In the past few years, Alexei Ratmansky choreographed a version based on the original Petipa/Ivanov notes and period costumes, and as far as I'm aware, it's nowhere near as popular as moden Nutcrackers or his other historical revivals. Apparently, it really did need to be heavily edited to become the classic it is now. Having danced it several times, there is something soothing about knowing exactly what you'll be doing around the holidays, and your family knows what they're going to see every time. This year has been wildly strange with most Nutcrackers cancelled. There's a lot of sadness and feelings of being at odds and ends from most dancers right now -- even the ones who hate breathing in glitter and soap flakes during every damn "Snow" scene every day, twice a day on weekends.
There's some scholarship out there on how Nutcracker turned into a Christmas phenomenon, and I cannot recommend them enough. In addition to Chan's excellent book mentioned above, check out:
Fisher, Jennifer. "Nutcracker" Nation: How an Old World Ballet Became a Christmas Tradition in the New World. New Haven, Conn: Yale University Press, 2004.
There's also a wealth of information about Petipa's ballets and how they transitioned out of Imperial Russia in Homans, Jennifer. Apollo's Angels: A History of Ballet. New York: Random House, 2011.