To really make the framing narrative of the thing work, Sholem Aleichem has to work hard to keep the text sounding all talk-y.
(Oh, quick brain snack, Shmoopsters: a framing narrative is what you call the outer shell when you get a story within a story, like here. The story within a story is all the stuff that Tevye is telling us. The framing narrative is the whole setup where we all agree to pretend like Tevye is saying everything to a guy named Sholem Aleichem that he keeps meeting up with.)
So how does he make it sound like speech? Well, he keeps the sentences short, not too complex, throws in a lot of colloquial language, makes Tevye actually address Sholem Aleichem by name, and pulls out the old trick of giving Tevye a verbal tic—in this case, the constant quotations. You can really see this at the beginning of each story. Check out this intro to "The Great Windfall":
If you are meant to receive a great windfall, do you hear, Pani Sholem Aleichem, it will fall right into your lap. As they say, it never rains but it pours. A stroke of good luck doesn't take brains or ability. But should it be the other way around—God forbid, you can talk until you are blue in the face, and it will do as much good as last winter's snow. The Talmud says: Without wisdom and a good idea—you might as well ride a dead horse. […] it's really worth your while to hear the whole story from beginning to end. Let's sit down here on the grass a bit. (2.1)
Okay, let's see what's going on here.
First, we've got all the details that make the text sound like it's an ongoing conversation. Tevye addresses "Pani Sholem Aleichem" and says that "it's worth your while to hear the story" to his interlocutor (ten dollar word alert! An interlocutor is the person you're talking to). Not only that, but he even gives the two of them stage directions in his speech, saying "let's sit down on the grass."
Second, we've got Tevye's very particular way of speaking—he's like a walking book of proverbs, aphorisms that use folk wisdom to make him sound, like, totally deep. Almost every sentence has some folk saying in it: "it never rain but it pours," "as good as last winter's snow," "ride a dead horse." And of course it wouldn't be our favorite dairyman without an allusion to the Bible—here, it's the italicized bit from the Talmud.