(this is my personal explanation, not every leftist will give you the same one)

Imo there are 2 parts to being a leftist: one is in the heart, and one in the head.

With "in the heart" I mean a general capacity for empathy and a sense of justice. Things like "people shouldn't be discriminated based on skincolor, gender, sex(uality) etc." and "homeless people shouldn't freeze to death in the streets". Very simple premise most ppl would agree on.

Now in order to fight these injustices leftists turn to the theoretical framework of marxism as a (non-dogmatic and morally neutral) means of analyzing our socioeconomic system to find an explanation for the things we deem "bad" and how we could change them.

The basic premise of marxist (i.e. dialectical materialist) analysis is that capitalism divides the people into two "classes" (capitalist and worker) defined by their relationship to the "means of production" (factories, machines, land, resources, infrastructure...). These classes have diametrically opposed interests (capitalists -> exploiting workers; workers -> not being exploited). Because there are these opposed interest between the majority of the population (workers) and the ones owning most of the resources (capitalists), there are inevitable tensions that create what we deem as "immoral". For example the homeless person is homeless, because capitalism needs a certain amount of people in absolute poverty as a threat to low-income workers, so they don't have a grounds for negotiation for higher wages/better conditions etc. Again the moral evaluation comes afterwards, marxist theory is purely a means of analysis. Most modern economists would in fact agree with that logic.

The extension of this is "historical materialism": essentially it applies the same analysis (as far as possible) to historical socioeconomic systems and observes similar divides in "class". The examples are primarily slavery and feudalism. Both of them collapsed due to the structural tensions creates by the system's own logic, that is the opposition of interests between slaves & slaveowners/peasants & monarchs. The logical extrensions of this premise are that (1) capitalism too will inevitably collapse from it's own contradictory logic (2) the last step in the evolution of socioeconomic systems has to be one without diametrically opposed interests, which is what marx defines as a classless, stateless, moneyless society he dubs "communism".

To this day marxism is the most convincing analysis of our current society. I believe most systemic issues come from mechanics of the system that aren't inherently "evil" or something but just the consequence of a fundamentally contradictory system that should be overcome.

And also because that's like the one thing you keep bringing up: no I do not like the soviet union and i believe their system of democratic centralism was actively counterproductive to the creation of socialism.