There are political, cultural, economic, and other reasons for the Soviet ascent in Chess and their continued dominance:
Succeeding in chess meant was attractive to the Soviet government because it was a way to be competitive in international sports. By defeating everyone else in chess, they were proving themselves to be superior intellectually.
By privileging group contributions over individual accomplishment, the Communist system was well-adapted to mobilizing to defeat opponents. Important GMs would have a team of ready masters working for them, helping them train, and develop their game. By contrast, most European and American players were fairly isolated.
This also changed the way chess was played. With a lot of people studying an researching openings and positions, the game got a lot of more theoretical, so it was harder for players in other countries to keep up.
A good bit of this was overseen by Botvinnik, who was not only an amazing world champion, but an organizational genius.
It was often in the schools and culturally valued. You could do it for a profession. In the U.S., by contrast, many of the top players (Reshevsky, Fine) had to work full-time jobs. Even if they were competing at the highest levels, they were almost amateurs by today's standards. They couldn't compete with people who studied the game 6-10 hours a day.
There were some exceptionally talented players and they got the chances they needed to succeed.
This is more controversial, and the extent of it is not known, but Soviet master reportedly threw or drew matches with each other to lock outsider players from winning tournaments. Fischer was really into this theory and I've seen people supporting and rejecting it.
I am sure there are more reasons, but I hope this gives a good overview of the multiple, interlocking causes for it.
If you're interested in a thorough treatment, I've given a long lecture on this (available on YouTube). I present the history, while my co-presenter, one of the best chess players in the United States, analyzes several chess games.
The story begins with Mikhail Chigorin (Wikipedia link). Chigorin stopped working to pursue chess full time. He published a magazine and played many informal and formal tournaments, culminating in a prestigious [edited] second-place finish at Hastings, 1895. This made Chigorin a household name in Russia and also attracted the attention of Tsar Nicolas II. After this, chess became a praiseworthy pastime (keep in mind many cultures had dis-favorable views of chess, viewing it as a waste of time or a potential vice due to gambling).
Tsar Nicolas II played an active role in establishing Russia as a venue for tournament chess, by sponsoring a junior championship (notable winner: Future world champion Alexander Alekhine) and designating the top 5 winners of a 1914 tournament in St. Petersburg as "Grandmasters of Chess".
World War I and the Russian Revolution certainly prevented chess tournaments for a few years, but Lenin himself was a fan of the game and even encouraged participation in chess. However, the biggest proponent was Nikolai Vasilyevich Krylenko (Supreme Commander in Chief of the Army). Krylenko famously proclaimed: "We must organize shock brigades of chessplayers and begin immediately a five-year plan for chess." (Soltis 25). Some described the Soviet attachment to chess as an ideological one. Yakov Rokhlin, a Bolshevik who knew Krylenko, said chess was "a true weapon and a living piece of propaganda against religious delusions." (ibid.) So the Soviet Union tasked the All-Union Central Council of Trade Unions (VTsSPS) with organizing large and small chess tournaments, and encouraging their members (most of the working population of the Soviet Union) to play chess. The Soviet Union set aside money to sponsor tournaments to attract top international players.
In an era where the average worker struggled, distinguished chess players used their influence to gain favor (sviazi). Chess players had help from powerful party officials (like Sergei Kirov) to obtain passports to travel internationally, to attend the best universities in the Soviet Union (Botvinnik obtained a doctorate in Electrical Engineering), and to receive a monthly stipend from the state (which could be yanked in case of disfavor--ask Mark Taimanov after he lost to Fischer in 1971).
Finally, I have to mention the crucial role of the Pioneers Palace (Wikipedia). Most every Soviet champion from Botvinnik onward learned chess as a child after school in a Pioneers Palace (Petrosian, Spassky, Tal). Even current Russian players got their start there (e.g. Peter Svidler). If you were talented and motivated, club directors would notice you and get you coaching and eventually pay for travel for participation in tournaments.
So the Soviet Union had an ideological justification for chess, a huge player base (making chess an integral part of the culture), a path to success through chess, and completely free training for top talent. After the fall of the Soviet Union, this system was shaken (Pioneers Palaces no longer exist, for example), but a strong cultural connection and extensive after-school programs still ensures that Russia has the most grandmasters in the world, by far.
The best book on this topic (and the one I've cited above) is:
Soltis, Andy. Soviet Chess, 1917-1991. Jefferson, NC: McFarland, 2000. Print.
Russia had a decent, but not spectacular, presence on the international chess scene before the revolution. Notable players in particular included Mikhail Chigorin, Eugene Znosko-Borovsky, Efim Bogolyubov, and Alexander Alekhine. Alekhine would go on to be world champion from 1926 to 1935 and again from 1937 to 1946. However, his loyalty to the Boshevik regime was dubious- he would become a naturalized French citizen in the 20s and was declared an enemy of the state by Nikolai Krylenko, who was the party official in charge of the chess establishment.
The impetus for state backing for the game came originally from Alexander Ilinzhenevski. He was in charge of training for military personnel in the post World War I period, and believed that chess cultivated attributes that would be positive for officers. He began organizing chess training within the military, and using his resources to organize chess tournaments open to the general public.
Eventually, Nikolai Krylenko would take over the administration of chess. Krylenko was a party official who had been a judge and prosecutor, and converted chess from a training method to a propaganda weapon- Soviet victories in chess would demonstrate intellectual superiority over capitalists. Under Krylenko, chess was elevated to the level of other sports, which under the Soviets meant nationwide recruiting of young talent and organized training.
Western countries, by comparison, do not generally promote chess or provide government support for training. Other Eastern Bloc countries provided similar support, but the Soviets had so many more people that the odds of producing top talent were much greater. The comparative structure explains a great deal, but let's talk about Botvinnik.
Mikhail Botvinnik was the first undisputedly Soviet world champion (reigned 1948-1957, 1958-1960, 1961-1963). He was loyal to the Soviet regime even through the worst excesses of Stalinism, and was promoted as the archetypal Soviet player. Botvinnik is widely regarded as the first true "professional" world champion. Previous world champions often were no more than talented amateurs and were not known to put in 40 hour weeks studying chess (as an example, Max Euwe, who was world champion from 1935-1937, was a full time professor of mathematics). Botvinnik would study his opponent's favorite openings and prepare surprises in those lines, often enlisting the help of a camp of assistants to invent new ideas.
To quote Kasparov (who admittedly may be biased as one of Botvinnik's students- "...a genuinely revolutionary leap was accomplished by Botvinnik...Botvinnik went further and created a system of opening preparation...Botvinnik deliberately disrupted the positional balance, choosing sharp, committal variations, but ones that he had studied deeply..."
In chess parlance, "sharp" refers to a position that is very double-edged, where even a very slight mistake is usually fatal. The advantage of superior opening knowledge is accentuated by sharper variations.
So, essentially, Soviet chess dominance can be credited to the combination of a support structure unrivaled by other countries, and a hard working, very talented champion who embraced a position as an archetypal Soviet hero.
Sources-
"My Great Predecessors, Volume III," Garry Kasparov, 2003.
"Storming Fortresses: A Political History of Chess in the Soviet Union, 1917-1948," Michael Hudson, 2013. Dissertation available at https://escholarship.org/uc/item/0s71f0cw#page-1
Generally, chess history sources are a little thin. The best English language chess historian is probably Edward Winter, who has a web site at chesshistory.com.