I watched it some time ago, but I still give it a shot (I hope my memory serves me well):
- Why is he eavesdropping?
He is socially isolated and has really no one, but his dog. At the end, after he went to Valentine's fashion show (his car also looked a bit dusty), she notes how it was good that he finally left his house. Also, when she visits him earlier in the movie, he reveals to her that it's his birthday (they drink pear brandy together) - no one else is there or seemingly cared, so he has to clink glasses with a then completely unknown person.
He has also become pretty cynical. The reason why she visits him is that she ran into his dog and injured it, but he doesn't care (well, he gives her (too) much money for the veterinarian). Also in their early conversations, their world views clash - she believes that all humans are good deep down, but he challenges her and clearly thinks otherwise.
It seems that he became used to eavesdropping and now listens to how other people live their lives for entertainment, since he has nothing else. It's also reassuring him in his pessimistic world view, e. g. when he challenges her by giving examples of how his neighbors treat each other (cheating, taking advantage etc.). And he calms himself by claiming that whether he eavesdrop or not, it won't change anything about their lives. His challenge to her (to reveal a husband's cheating) is also pretty unfair.
Roger Ebert described it thusly:
He occupies his days intercepting the telephone calls of his neighbors, and he watches them through his windows almost like God--actually, just like God--curious, since they have free will, what they will do next. After a lifetime of passing verdicts, he wants to be a detached observer.
- Why does he stop?
I'm sure that he doesn't consider her to be someone unknown any longer. After all, they have met several times and he has learned a lot about her and also told her about himself. She is also the only human he meets face-to-face in his life. And while eavesdropping his neighbors leaves him somewhat detached from them, when he talks to her, he sees her vulnerability, her emotions. He experiences what that means to the people involved - when he brings up her drug-addicted brother and she quickly leaves, but he notices how she was crying.
All the time before, he was in his world and his neighbors were in theirs (they didn't know about his spying), but Valentine bridges these two worlds. She is not just a voice on a recording, or a line in a newspaper, but an actual human being. And she is afffected by it. After he has hurt her and when they meet again, he tells her how he saw that she was crying and that he believes she was disgusted at him - and he tells her that this belief prompted him to report himself.
According to Kieślowski
“the essential question the film asks is: Is it possible to repair a mistake that was committed somewhere high above?”
The judge acts like God in the beginning, but with Valentine he slowly reenters our world. This is more thoroughly described by Norman Holland and he says:
By the end of the film, the judge has been redeemed, changed from a man with nothing but angry contempt for his neighbors to someone who empathizes with them, someone who can love.
There is probably not a clear answer as to why he report himself, but I'm sure he did it for himself and her. She was disgusted at him and he now cares about her, that makes sense. But maybe he also understood that it was important for himsel to report his crime and face the consequences. This way, he can put an end to his earlier, after all, miserable life.
To end again with a quote from Kieślowski:
There’s something beautiful in the fact that we can give something of ourselves. But if it turns out that, while giving of ourselves, we are doing so in order to have a better opinion of ourselves, then immediately there’s a blemish on this beauty. Is this beauty pure? Or is it always a little marred? That’s the question the film asks. We don’t know the answer, nor do we want to know it. We’re simply reflecting on the question once again.