This one's easy: every story is told to Sholem Aleichem (and, so, to us) by Tevye himself and about himself. Other characters show up, sure, but only to illustrate some point that Tevye's making. Bo-ring.
Or is it? We tend to associate first-person narratives with autobiography, but this one is a little different. It's not a work like Great Expectations or The Good Soldier, where the person is telling a story after the events are over.
Instead, Tevye is more like a journalist reporting on himself. He doesn't really make the connections between stories or craft a meaningful narrative to explain his life. One super-duper crazy example? This little passage from the "Hodl" story, when Hodl tells Tevye that she is going away to be with Perchik in his exile:
I will never forget the way she looked at me. I thought she meant she was going to drown herself. Why? Recently, may it not happen to anyone, a girl living not far from us fell in love with a village Gentile, and because of him—well, you know what happened. […] The village Gentile thought it over and decided to go off with someone else. The girl then went to the river, threw herself in, and drowned herself. (5.124)
Why is this so majorly freaky? Well, friends, it's because these are the very same things that will happen to Tevye's other daughters! Chava will indeed ruin her life by running off with a village Gentile, and Shprintze will drown herself when her love affair doesn't work out. The first time you read this, whatevs, no big deal. But in hindsight?... Chills.
So why are the stories written like this? Well, for one thing, Sholem Aleichem writes as though Tevye hasn't really had time to reflect on the most recent events, and he tells them to us in the breathless, can-you-believe-this way of someone who is still experiencing the emotions he's describing.
For another, because he doesn't really have a lot of distance from the events he's talking about, Tevye tends to react to them all the same way—by doing that thing where he acts like a big martyr and wonders why God is just always out to get him.
Or—try this out. Remember that Sholem Aleichem kept getting confused about how many kids Tevye was supposed to have? (Check out the plot summary of "Hodl" for a quick refresher.) Maybe he really didn't know what was going to happen to Chava and Shprintze yet.