The secret section of the Soviet-German non-aggression pact had assigned eastern European states to spheres of influence. Finland, being both a former constituent of the Russian empire and directly on the soviet border, was agreed to be a part of the soviet sphere. Latvia, Estonia, and later Lithuania also belonged to the soviet sphere and under military threat had signed assistance agreements with the USSR.

Finland was important to the soviets strategically. They were in possession of Karelia, and as such an invasion launched from Finnish territory would leave Leningrad extremely vulnerable and without any buffer territory. It was therefore obvious to all party members that Finnish territory must be acquired, in case a foreign power would want to use it for access into soviet territory.

A Finnish delegation was received in Moscow and given a draft of an assistance pact, which would have guaranteed military access for the soviets as well as territorial concessions, in the form of Karelia and the strategic northern ports of Petsamo. This was rejected on account of the Finns needing the approval of a parliamentary majority to be able to accept soviet terms. Even during this meeting, Molotov threatened military action.

So the more important party members already saw the invasion as an absolute necessity. It was simply a matter of survival to them. However, I assume your question is more about the general population. So, on to answering that. Prior to the invasion of Finland, the soviets had several border incidents that devolved into local skirmishes. These, of course, were portrayed as proof that the imperialist powers were threatening the integrity of the USSR. In both cases, these incidents had been instigated by the japanese and soviet response was largely justified. So you have a massive propaganda machine telling people that the country is surrounded by enemies seeking to undermine its stability and territorial integrity, and an atmosphere of public paranoia and war readiness. This made it all the more credible when, directly prior to the outbreak of the winter war, soviet radio announced that there had been a provocation on the Finnish border, and soviet military response was required.

So this is how the invasion itself was justified. However, propaganda for the war did not end there. Soviet media was notorious for how unrealistically pervasive propaganda was, and this is no exception: a revolutionary government, in reality a soviet puppet, was set up under the leadership of exiled Finnish communist leader Otto Kuusinen. Soviet propaganda, in response to this, began stating that the soviet military involvement was merely spreading the revolution, that their fight was for the liberation of the Finnish proletariat.

Hope this answers your question.

Sources:

Ziemke - the red army 1918-1945

Montefiore - Court of the red tsar

Memoirs of Nikita Khrushchev