This is an ellipsis: "…" It comes into novels to show a hesitation or a pause in dialogue, or to indicate when something is being deliberately left out of the narration. So, for example, let’s look at this sentence: “Yeah! You look … great … in spandex pants.” All those little dots suggest that this person is telling a big ol’ lie: she’s obviously hesitating a lot.
There are two moments in Anna Karenina when we get several lines' worth of dots. These dots show us, as in the example above, that the narrator is hesitating to say something, or he's leaving something out. The first comes at the end of Part 2, Chapter 10, when Anna and Vronsky consummate their affair. There's no description of the sex, but the dots stand in where it would have gone, if Tolstoy were a different kind of author.
The second time comes in Part 5, Chapter 23, at the end of a sentence that starts, "The doctor told me after my illness …" That's the moment when Anna's telling Dolly that she won't be having any more kids. And again, whatever the doctor actually told Anna is left unclear, but the ellipses make it obvious that something missing. In both cases, all those dots draw attention to things that are so terrible, and morally damning, that the narrator can't bear to say whatever it is. But by not saying something and leaving in the dots the narrator is making it obvious exactly how serious he thinks what he's not saying is.