First off, one has to realize that although historians sometimes talk about "the Soviet period" as one single era (I myself am guilty of this), Soviet history was far from a monolithic bloc and the Soviet cultural scene could vary quite widely between the 1920s and the 1980s. So while troupes like the Bolshoi performed throughout the existence of the Soviet Union, what they performed could vary from experimental dance to a more traditional ballet and all sorts of dances in between.

Although the Bolsheviks initially toyed with the idea of disestablishing the ballet troupes as a legacy of bourgeois arts, they elected to keep the troupes under state pay. The 1920s ballet was typified by experiments trying to create a new Soviet culture and a Soviet man. Socialist-realist ballets like The Red Poppy in which Russian sailors act as the protectors of downtrodden Chinese were a departure from the traditional ballet canon. Ballet under Stalin became torn between a socialist-realist dance forms and older conventional dance. The latter was somewhat popular among both the Soviet public and the Stalinist state which prioritized kulturnost (culturedness, a colleague of mine said the best English translation is "a gentleman and a scholar"). One of the important ideas that animated enthusiasm for the Stalinist state was that it was working to bring high culture to the everyday man. This prioritization of culture became more acute during the Cold War as the Soviet state saw its promotion of high culture as a counterweight to America's crass mass consumer culture. Added to this trend was that during the Great Patriotic War, the Soviets tended to rediscover Russian culture and Russian nationalism, albeit modified to fit the Soviet case.

Ballet fit within this cultural matrix because it could be simultaneously a symbol for how the Soviet Union had transformed the archaic old order or be an example of how the state preserved high culture and brought it to the masses.

Sources

Ezrahi, Christina. Swans of the Kremlin: Ballet and Power in Soviet Russia. Pittsburgh: University of Pittsburgh Press, 2012.

Fitzpatrick, Sheila. Everyday Stalinism: Ordinary Life in Extraordinary Times : Soviet Russia in the 1930s. New York: Oxford University Press, 1999.

The Moscow Ballet is a very famous company, and the Soviet Union was pretty restrictive about art, but ballet was never the only permitted art form or anything like that. If you google "Soviet art," you'll find painters, writers, animators, musicians, architects.. Any contemporary art form you can find outside the Soviet Union, you could find within it.