I would say that it is a bit broader than that.

On a very basic level, the Cold War was a contest of ideologies. In one camp you had the Soviet Union (and others) which embodied an ideology of Marxism-Leninism; featuring aspects like state bureaucratic control over economic planning and state ownership of enterprises, one party elections and one party rule, strict controls over information, routine surveillance of citizens by the state, etc.

In the other camp you had the US and Western European allies. This camp (generally) espoused an ideology of liberal democracy and capitalism. Meaning, these countries tended to feature competitive multi-party elections with universal or near-universal suffrage, free press and freedom of citizens to express political opinions, private control over enterprises and limited state intervention into the economy.

So, when the Eastern European regimes and the Soviet Union collapsed and were succeeded by states that embraced capitalism and liberal democracy, there arose a feeling in some intellectual circles that capitalism and liberal democracy had "won" the ideological contest and Marxism-Leninism had "lost". As an example, Francis Fukuyama wrote The End of History and the Last Man in 1993, where he predicted that humanity was entering a period where slowly-but-surely all governments would come to resemble the Liberal Democracies of Western Europe and the US.

That is what I understand Cold War Triumphalism to be. The belief that because the Soviet Union fell, ergo Communism is discredited, ergo Liberal Democracy is the only viable model for governance in the world. It's also the belief that populations around the world that are not living in liberal democracies want their countries to be liberal democracies. In this line of thought, the only thing that prevents the people's natural desire for liberal democracy is resistance or repression by their illiberal or authoritarian governments.

Said another way, Cold War triumphalism is the belief that the US (and perhaps also Western European countries) have figured out the best way to live and govern themselves, and that all the other people of the world want to be like the US. And also that the US and Western Europe should do things to help other countries be more like them, by stating support and solidarity with protest movements, or through military pressure and economic sanctions against illiberal and authoritarian regimes to encourage regime change.

That line of thought runs into problems in practice, because it makes essentially no allowance for states like China, Vietnam, Ethiopia, or Kazakhstan. That is to say, countries that are "developmental states" which purport to make an implicit bargain with citizens of limited political freedoms in exchange for robust economic growth and visible improvement in overall quality of life.

Edit- decided "liberal democracy" doesn't need to be capitalized