Conjunction Function
Tolstoy's style seems to have a crush on coordinating conjunctions—you know, and, but, or and so on. If you look at any extended passage in Anna Karenina, chances are you'll see piles of sentences starting with these conjunctions. This type of sentence structure indicates that whatever comes before the conjunction is both connected to and just as important as what comes after.
The effect this has is to make Tolstoy's long paragraphs like logical accumulations of ideas: piles on piles of sentences that create a total portrait of a character's inner thoughts. By throwing lots of ifs, ands, and buts at us, Tolstoy's style gives us the impression of completeness. Each paragraph is like a perfectly self-contained house, with early sentences providing perfect foundations for later ideas and observations.
Just check out this super-insightful look into Karenin's psyche (it starts with a conjunction!):
But as time went on, he saw more and more distinctly that however natural the position now seemed to him, he would not long be allowed to remain in it. He felt that besides the blessed spiritual force controlling his soul, there was another, a brutal force, as powerful, or more powerful, which controlled his life, and that this force would not allow him that humble peace he longed for. He felt that everyone was looking at him with inquiring wonder, that he was not understood, and that something was expected of him. Above all, he felt the instability and unnaturalness of his relations with his wife. (4.19.3)
Just as Tolstoy's larger plot ideas flow logically from conflict to complication to solution, on the sentence level, his words move from one sentence to the next smoothly and easily, as though nothing in the novel could be phrased any other way.