I've been involved with solving some mysteries and I think your question is almost too broad.* Every case is a little different, and the impact they have varies accordingly. I'd say my most common feeling, for cold cases at least, is kind of relief mixed with sadness, if that makes sense.

But even that varies. I once felt pure elation after my dog found some remains. My dog was newly certified and the remains were 6 years old, and she'd never trained on remains that old but still did a textbook perfect alert. I felt like a proud parent until later when I was giving statements and filling out paperwork and had time to really think about the family.

I've also just felt deep sadness sometimes, but, somewhat narcissistically, I have to admit that these are usually victims who remind me of myself, so I identify with them more.

Reading about cases online is interesting but I don't get emotionally involved, so I guess when I find an explanation that I accept, it's sort of satisfaction but not really that strong.

*I don't mean this as a criticism, it's an interesting question