It was part of the resurgence of nationalism in the late years of the USSR leading up to the Union's dissolution. Starting with Gorbachev's glasnost and perestroika reforms, dissent was slowly permitted to return after decades of crackdown by the Soviet government. A major part of the Soviet Union's dissolution was due to the various member states beginning to more fully express their own nationalism and independence. As part of that process, a natural byproduct is a rejection of Soviet ideals.

The reasoning behind the rejection of Soviet tenets and ideological beliefs isn't as complicated as you might think. Remember, the Soviet Union was a "union" in name only - in practice, the Union existed to strengthen and serve the Russian Soviet Federative Socialist Republic, by far the largest, strongest, wealthiest, and most populated Soviet republic.

The Soviet government's policy of exploitation within their "sphere of influence" led to events such as the forced relocation of people from their home countries to other parts of the Soviet Union depending on what the "needs of the country" were. For example, the Soviet government is building a city specifically to mine nickle and other ores in the far north and they need workers - the Soviet government would order the population in a certain area to relocate to the new city under threat of imprisonment. This type of policy led to the mass depopulation of Kazakhstan, the Caucasus, parts of Ukraine, and other Soviet member states.

Policies like that tend to breed a lot of resentment towards the government. Focusing more on your question regarding the abandonment of socialism/communism, the Soviet implementation of a command economy certainly impacted the vast majority of everyday citizens, especially during the stagnation period in the 1970s and 1980s under Leonid Brezhnev and his successors, Yuri Andropov and Konstantin Chernenko. Even as far back as Stalin's collectivization policies in the 1930s (which led to a manmade famine in Ukraine called the Holodomor where millions starved to death when the government confiscated Ukrainian grain to distribute across the entire Soviet Union), the Soviet economy was directly manipulated by government-mandated production quotas, price fixing, and industrialization goals which were rarely, if ever, fulfilled. While there were some benefits to Stalin's (in particular) direction of the command economy with regards to industrializing the rural, backwards Soviet Union (in Stalin's years as Secretary General, the Soviet Union went from being vastly agrarian to a nuclear power, all within 30 years), it tended to cause far more problems than it solved. A good example of this is the economy under Brezhnev:

Brezhnev's regime saw chronic shortages of almost every good you could think of, from vehicles to basic staples like bread. To illustrate this, a couple of stereotypically cynical Soviet jokes you would've heard in this time:

We pretend to work and they pretend to pay us

What is more useful, newspapers or television? Newspapers, of course! You can't wipe your ass with a television!

A man walks into a shop and asks, "You wouldn't happen to have any fish here, would you?" The shopkeeper responds, "You've got it wrong - this is a butcher shop. We wouldn't happen to have meat. The fish shop is across the street - THERE, they wouldn't happen to have fish."

Having experienced chronic goods shortages, poor quality of the goods that were available, and inflation caused by the Soviet command economy in conjunction with the repression, censorship, and authoritarian abuses by the government and its agencies, the former Soviet member states outright rejected socialism/communism as part of their general rejection of Soviet ideology in favor of trying something new.

Sorry for the wall of text, but I hope this was able to answer some of your questions!