We all have a stake in keeping the sites clean, not just moderators. Anyone who posts regularly on Stack Exchange or even just frequently goes to Stack Exchange for information has a vested interest in helping to keep the site clean so that they don't have to wade through piles of garbage to find stuff to answer or the information they're looking for.
I remember the "bad old days" before Stack Overflow where when you had a question you had to wade through discussion forums that were filled with clarifying questions, "me too!" answers, and thread hijacking just to find the one piece of information you need in order to complete your task.
I have a strong interest in making sure that that doesn't happen on Stack Overflow, so whenever I see this kind of answer I flag it, and I work the Very Low Quality queue to help delete this kind of content. This helps make sure that I can quickly find the information I need when I'm looking for it. I use the site to find information too (I can't even count how many times Stack Overflow has helped me get "unstuck" on a tough problem or find a weirdly specific piece of information I was looking for), so it's just as much my responsibility to keep it clean as it is anyone else's.
One of the things I appreciate about Stack Overflow and the other SE sites is that they're focused on answers, not just people sharing opinions. If you're looking for concrete information, it's much easier to find it here than it is on old discussion forums. While entertaining to participants, most people simply don't care about the records of other people's discussions on the topic - they'd rather have a site where they can quickly find what they need.
I've noticed on a couple of sites that moderators aren't moderating their sites. They seem to be waiting for high-rep users to do it.
The entire point of giving high-rep users these privileges in the first place is so that they can take care of moderation tasks that don't require moderator intervention.
This is particularly the case on sites like Stack Overflow, where in practice there is far too much needed moderation for the current moderator staff to take care of themselves.
Stack Exchange sites are community-moderated, not moderator-moderated. It's everyone's responsibility to help keep the site clean.
Aren't moderators supposed to show users how each site works, keep questions clean, edit bad titles, give askers tips etc.?
No, that's what high-rep users are for. (Not to mention each site's site tour, FAQ, and Help Center; there's plenty of documentation available on how the sites work, it's just that many new users never read it).
I always thought that high-rep users with additional privileges only support moderators in their work but it looks like they're actually the ones doing it.
No, high-rep users do things that the community's allowed to do, and moderators do things that the community's not allowed to do. Moderators do things like moderate comments, track down sock puppets and voting fraud, etc. that the community can't do itself.