Let me start with links to a couple previous answers I've written on the subject:
So why did liberal democracy (for all the corruption and serious strains it is currently under) "catch" on more in countries like Poland or Hungary than in Russia? I'll summarize a few differences here.
From a very early point in their transition, former Eastern Bloc countries had an eye on integration into European institutions like the EU and NATO, and this acted as a carrot to stick through with market reforms and political reforms. Slovakia was very prominently denied a NATO invitation in 1999 (unlike Poland, Hungary and the Czech Republic) because Vladimír Mečiar's government was considered too autocratic and corrupt, for example. Russia never was considered/considered for itself membership in these institutions in similar terms as these other countries.
Other Eastern Bloc countries (including the Baltic states) also could frame their postcommunist transitions in terms of getting rid of foreign occupiers and collaborators with foreign occupiers. Russia (and most of the other former SSRs) not only didn't have this option to just make huge chunks of the former system leave and retire, but they actually had to absorb those elements that were being removed from Eastern and Central Europe.
Similarly, Russia didn't have a stable pre-communist order that it could point to and try to regain, only an extremely turbulent period between the February and November 1917 revolutions. In contrast, Eastern and Central European countries had much more recent models they could work with. On top of this, most of the European "People's Republics" never had quite progressed as far along in Marxist-Leninist socialism as the USSR had - most of them had still had token non-socialist parties (the smaller parties finally voting independently and breaking with the ruling party in Poland was a big step in ending communist rule in 1989) and/or independent sectors of the economy and of society. Poland (and Hungary and Lithuania) again had the massive presence of the Catholic Church (which wasn't under communist control), and had never collectivized agriculture.
Because of the nature of the Russian constitutional order and the protracted Soviet-then-Russian constitutional crisis, as I discuss in that second linked answer, Yeltsin ultimately consolidated his power by making alliances with/gaining control of the existing security services and governmental institutions. While there was a Democratic Russia movement in 1991 that was modeled roughly after Polish Solidarity, Czechoslovak Civic Action, or Lithuanian Sajudis, it never was given the chance to contest open national elections. There's a big "what if" if Yeltsin had called elections in the autumn of 1991, but for a variety of reasons he didn't and so Russia kept its Soviet legislature until it was forcibly disbanded in October 1993, and by this point the Russian electorate had soured on the liberal reformers associated with the Yeltsin presidency, like Yegor Gaidar. The combined parties that were either affiliated with Gaidar or had their origins in the Democratic Russia movement scraped together about 30% of the vote in 1993, with the far right Liberal Democrats getting 22% and the Communists and their Agrarian allies getting 20%. In the 1995 elections, with worsening economic conditions and the increasing popularity of Yeltsin, these groups declined even further, with the Communists becoming the largest parliamentary party and making a serious bid for the Russian presidency in 1996. Also by this point Yeltsin himself had largely abandoned the liberal political and economic reformers, and developed a "party of power" of former Soviet officials, such as his long-serving prime minister Viktor Chernomyrdin, who had been the Soviet Natural Gas Minister and then head of Gazprom. Yeltsin, physically weakened in his second term, very much relied on oligarchs for his continued rule - grassroots democratic organizations had effectively ceased to exist by that point in the political and economic chaos.