The short answer is the game was not changed, but the players were.

The popularity of chess in the Russian Empire surged with Mikhail Chigorin (1850-1908), who challenged Emanuel Lasker William Steinitz for the World Championship in 1889 and 1892, losing both matches. During this time, there was really only match play in the Russian Empire (no tournaments). This changed in 1899, when the first All-Russian Master's Tournament was organized (Chigorin won). These championships still occur almost annually, though the name and format has changed a few times to reflect government changes.

Chigorin's success inspired other players, like Akiva Rubinstein and later Alexander Alekhine. These two players would go on to win several Russian championships, and Alekhine would go on to be world champion for many years. Related to your question, chess during this time was practiced primarily by the aristocracy. Alekhine's father, for example, was the governor of a region and member of the Duma. Chess tournaments were sponsored by wealthy patrons, like Tsar Nicolas II, who paid for the prize pool of the famous 1914 St. Petersburg tournament.

The Russian Revolution changed this dynamic, as it largely abolished the aristocracy. However, Lenin was an amateur chess player and fan of the game. He tasked Nikolai Krylenko with developing chess in the RSFSR (later USSR, kinda). Krylenko established "shock brigades" to bring chess to the masses, by sponsoring tournaments in factories, chess leagues in cities, inter-city tournaments, etc. Later, this became formally organized by the State Committee for Physical Culture and Sport. This committee decided which players attended which tournaments, which players received stipends to support their development, which players would be tasked with coaching, etc.

The player profile also changed. The central committee favored the promotion of working class players of pure Russian (non-Jewish) roots, like Vasily Smyslov. The rules of chess (including the names of the pieces) were not changed by the Russian Revolution, but the context was. One fine example of this transformation can be seen in the Anichkov Palace in St. Petersburg. The residence of many nobles, including a young Tsar Nicholas II, the palace was converted into a Young Pioneer's Palace, housing after school clubs like chess. Many of the USSR's strongest players literally learned chess in a former Tsar's (Alexander III) study.

The best book on this topic is Soviet Chess 1917-1991, by Andy Soltis.