Andrew Soltis' Soviet Chess, 1917-1991 is the book to read. I put my library's copy back a little while ago, so I don't have it in front of me, but I can try to summarize the relevant parts for you. If it's too general, I can get the book tomorrow and attempt a more proper post.
After the Russian revolution there was a big push to bring all of Russian life and culture into line with Soviet ideology. Artists, writers, and other creatives and intellectuals could either conform and produce work that supported the ideals of the revolution or be punished, and punishment could go so far as imprisonment in a gulag or even execution. The abstract nature of chess saved it from many of the purges that afflicted other intellectual pursuits. Pro- or counter-revolutionary ideals just don't fit into chess play, but as a vehicle to promote the image of the Soviet citizen as a logical, sophisticated being, it was ideal. Some players did get caught up in party politics and rose and fell to the fortune and intrigue that accompany it, but by and large chess and chess clubs were allowed to flourish. It certainly helped that Lenin was seen to have played chess from time to time.
The 1972 Reykjavik match is what everyone thinks of when discussing East vs West on the chessboard, but this rivalry started very early on in the Cold War. In 1946 there was a chess tournament between the American and Soviet militaries stationed in Berlin. The Soviets roundly defeated the American contestants, and remained dominant in the chess scene until, and then again after, Bobby Fischer. Even before the Berlin match, though, chess schools had been established for Russian players.
That, though, is where my memory starts to fade. I'll grab a copy of Soltis' book and of the tournament bulletin for Berlin '46 tomorrow and provide more detailed citations.
Hi, we've had some threads on chess which may be of interest
How did the Soviets come to be so dominant at Chess? featuring /u/rdC19, /u/JediLibrarian and /u/iattorn
Why were Russian chess players so successful in the Soviet era? featuring /u/nothingbutjelly, and links to a few other posts
Why Were Chess Players in the 60s-70s Treated like Celebrities and Not in Todays Society? a quick comment from /u/dar212
These threads are archived, so if you have questions for any of the users, just ask here & tag their username to notify them