Actually USSR pushed a lot of soft power and spent real money behind it. Especially the authors whose narratives didn't directly violate the narrative of the Party.

Leftist parts of society looked up to USSR a lot, and a lot of humanities professors, teachers all over the world were left-leaning, and promoted these books as Russian culture.

This is one factor, and doesn't explain the whole thing, of course.

See: https://news.ycombinator.com/item?id=48673777. Neither our family nor I ever leaned towards the Party or any form of Leftism, but books are always kosher in our culture.

Well it is kind of an integral part of the classic european literature canon. Certain novel genre's were invented by Russian writers, say, Tolstoy's epic novels.

So the question really is how does one find out about classic writing overall? Outside of school?

Well, Crime and Punishment is often taught in US high schools. Or at least was back in the 1980s. During the Cold War there was a lot of interest in Russian literature in general in part because in the bipolar world they were seen as "the other civilization" that we needed to understand if we wanted to avoid nuclear war. You'd think these days people would be more into Chinese literature but except for the Three Body Problem series, I haven't really heard of many Chinese books becoming popular here.

The same was as literature from anywhere else. Some authors are famous and you grow up knowing about them: I know about great Russian authors the same way I know about great authors from anywhere. They also get referenced and quoted by authors in other languages. Playwrights get translated and performed.

Less famous authors? Everything you say and more - again, just like any other books and author.

I was a precocious teen - reading philosophy and history in addition to fantasy and science fiction - and came across Crime & Punishment in a local bookstore that I purchased paperbacks and Dungeons & Dragons books. The back sounded interesting and the book looked deep and philosophical so I purchased it. Despite it not being a great translation I found the book and topics covered really interesting and went on to read most of Dostoyevsky works.

This is just the past and the modern propaganda nothing more.

those books are considered as classics because they deal with different aspects of the human condition which haven't changed significantly. they also give a different and valid intepretation / perspective on these "eternal topics", which are unique and discernible enough from their counterparts from other countries and cultures.