I think that there could be many reasons why Russians support a special operation.
In glorious Soviet times Russians ruled 16 countries (including Ukraine) that were parts of USSR, had puppet governments in Eastern Europe and friendly governments in Middle Asia and Africa (and by the way Russian Empire was even larger and included territories of modern Poland and Finland). Traitor Gorbachev surrended Eastern Germany and East European countries to America, and traitor Yeltsin (first President of Russia) helped to disintegrate USSR. Many ex-USSR countries were lured into NATO and turned against Russia. Due to Yeltsin following malignant American advices, Russian economy has collapsed, crime rate has raised, and oligarchs have stolen valuable assets, leaving people in poverty.
America also defeated many former Russian allies, like Yugoslavia. Then America refused to stop NATO expansion to the East.
So some people might want a revanche. They want Russia to be strong, victorous and glorious and not a third-world country surrounded by American bases preparing to make a final blow. And if that is not possible, at least they want to stop NATO from expanding to the East.
Also, in Ukraine there are Russian people and some believe that they are treated there unfairly. For example, they are not taught Russian language at school, there is no Russian television and so on. Obviously, Russia must protect them.
Also, propaganda doesn't call it a "war". It is actually illegal to call it a war. It is a "special military operation" for liberation of Ukranian people from Nazis and American influence. As Putin stated, Russia doesn't plan to occupy Ukranian territories.
Russia had no choice and no other means to prevent a big war planned by the West, propaganda says.
So there might be people who believe propaganda. If they have no other sources of information, what should they think?
Also, here are some quotes from a pro-government news agency TASS. Hope they give an idea how propaganda portrays the situation around Ukraine:
> the majority of Russian respondents - 88% - agreed that there are nazist organizations in Ukraine that are a threat to Russia ... 70% of respondents stated that Unkrainian government supports such nazist organizations ... According to the results of the poll, most respondents expect that special military operation will end with a trial of Ukranian nazis for commited crimes. [1]
> Medinsky (presidential aide) : The crimes commited by Ukranian government against Russian people in Ukraine became a challenge for Russia ... Russia has to protect peaceful civilians, its compatriots, who were assaulted, killed, threatened... [2]
> Russian senator Ekaterina Altabaeva: one cannot exclude Russia from world politics, and current special operation for denazification of thе country and protection of Russian-speaking citizens affirms a fateful role that Russia has always played in the world history ... at the end of 18th century not a single gun in Europe was allowed to fire without Russia's permission... [3]
And here is quote from other news agency:
> Military operation in Donbass is not a beginning of a war, but an attempt to prevent a global war, said spokeswoman of Ministry of Foreign Affairs ... also, it is an ending of the war that has been continuing for 8 years. [4]
So if someone wants to understand what an average Russian sees in the news or in social networks or on the Yandex frontpage, try reading sites like TASS or Interfax Russia, or 1tv.
[1] https://tass.ru/obschestvo/14412735
[2] https://tass.ru/politika/14414033
[3] https://tass.ru/politika/14412115
[4] https://www.interfax.ru/russia/824241