You know what Tevye the Dairyman totally sounds like? Like one of those talking head to-the-camera confessionals on reality TV shows. You know, where they take a cast member and sit him/her down in front of the camera to comment on the action that's happening in a funny or mean or revealing way. Think about it.
First, Tevye is telling us what has happened to him pretty much right after as it's happening. That's why he doesn't always seem to get the connection between, say, telling a story about a girl drowning herself after her fiancé deserts her and what happens to his very own daughter.
Second, so much of what he says is a big old confession about the feelings he usually is desperate to conceal. Check out the difference between what actually happened when he saw his daughter Chava after she has married Chvedka, and the feelings he lets us in on:
I lifted my eyes and looked—Chava! She was the same Chava as before, not changed by a hair, even wearing the same clothes! My first impulse was to jump off the wagon, take her in my arms, and kiss her, but a thought held me back: Tevye, what are you, a woman? So I pulled on the reins […] something tore at my insides and tugged at my heart. I was about to jump off the wagon, but I restrained myself and pulled the horse to the left. […] you must know how much I wanted to turn and look back at the spot where she was standing. But no, Tevye is not a woman. Tevye knows how to conduct himself before Satan the Tempter. (6.103-105)
Outwardly, this is totally harsh. Just picture him riding away from a crying, screaming daughter without even acknowledging her existence. Seriously cold behavior, right?
But here, because of the way he tells the story, we get to see that he's ripped apart on the inside by feelings that are struggling against his sense of pride and betrayal. It's a private confession that—oops!—the whole world now gets to read.