When I think of the early days of modern chess (say, the 19th century) I don't think of players from any particular country or region, just generally around the world, probably mostly European. However, somehow somewhere between its founding and the 1960s, the Soviet Union came to dominate the rest of the world, at least in my extremely sketchy grasp of chess history.
I'm especially thinking of things like the Fischer-Spassky match in 1972, which was a huge public spectacle seen by many as a sort of proxy for the Cold War itself: a lone brilliant American staring down the Soviet machine. But as far as I understand, prior to that chess was all but nonexistent in the mind of the general public in the US, and that was very much not the case in the USSR. How and why did chess become so entrenched (priveleged?) in Soviet culture? Was it that way in pre-Soviet Russia too and I just don't know about that? There doesn't seem to be anything about chess in general that might make it especially appealing to the Russian people, but on the other hand things don't typically emerge as national pastimes without some story behind them...
My comment which /u/Georgy_K_Zhukov referenced does a decent job explaining why the USSR became so dominant at chess. But I do want to clarify one point from your comment:
But as far as I understand, prior to that chess was all but nonexistent in the mind of the general public in the US...
The United States had a well-developed chess scene, but it was largely confined to New York City at this point in time (early-mid 20th century). Recall that the United States had it's share of great chess players before Fischer, from Paul Morphy to Harry Pillsbury to Frank Marshall to Reubin Fine (and I know I'm glossing over Reshevsky, Paulsen, et al). However, nearly all of those players considered chess secondary. Morphy abandoned chess when he encountered disapproval from his family and potential romantic partners. Fine made chess secondary to a very successful career as a psychologist. Pillsbury was struck down in his prime by syphilis. Arguably, of the early American champions, only Marshall qualified as a chess professional--until Fischer.
Chess has never been as popular in the United States as it is in Russia. Americans of a certain generation nearly all knew the names of Babe Ruth, Mickey Mantle, Wilt Chamberlain, Mary Lou Retton, or Tiger Woods. But chess was the national pastime of The USSR, and there, national heroes included Botvinnik, Petrosian, Keres, Smyslov, Tal, et al.
I establish in my linked answer the political will behind making chess their national pastime, but it's also worth mentioning other aspects making chess a pervasive part of Soviet culture. Here are a few factors most do not consider:
Chess is cheap to play. Any beginner nowadays can get started for less than $10, and that investment will last you years. If you look at where chess is growing the fastest in the United States right now, it's probably the border region of Texas, in part because it's much cheaper to play chess (or soccer) than it is to play football.
Chess requires little to no language. In a country as diverse linguistically as the USSR, chess is universal.
Chess is egalitarian. In a sport like golf, expensive equipment garners an advantage. As compared to a card games, there is no element of luck in chess. You win or lose on your own merits.
Chess fosters collaboration. Players require an opponent, and good players require a teacher (or many teachers). Players can almost always learn from one another, even with a difference in ability. In the USSR, strong players were encouraged to transfer their knowledge to the next generation, epitomized by Botvinnik's famous chess school (where Kasparov and many others learned).
More to be said, certainly, but this thread got multiple responses for a similar question, with nods to /u/JediLibrarian and /u/iattorn.