I've been told everywhere (including the help center!) about the fact that Stack Exchange is run by the community. Why is this?
Well... Who else would run it?
The staff is pretty small, so expecting them to do everything on all ~170 different sites would be a big ask. Alternately expecting them to increase the staff so that they could have enough people on staff to do it would be extremely expensive. There are over 500 volunteer moderators on the network, moderating between one and seven sites each. While they're obviously not full-time, replacing them would require a good number of staff.
But that's only a very minor part. Most of the actual "running" of a site is done by normal users like you... Users who have attained privileges due to the reputation they have earned by asking and answering questions.
That's millions of people who are editing, closing, deleting, flagging... and many other tasks involved with "running" the sites.
It'd be impossible to do it any other way and still have the same level of quality that we have now. And the users seem to like that degree of control. We enjoy being able to help improve our communities and being self-sufficient is a reward. It grants a feeling of ownership and community that just answering or asking and voting would create. This is good!
Stack Exchange is run by Stack Exchange Inc in the USA. But they give a significant freedom of your votes and a relatively lenient moderation mechanism.
The system is clearly tuned to motivate you to contribute as much good content as you can. In exchange, you have an above average protection from the flood of crap everywhere on the Internet, and from the individual childish power misuses of the "good old boys" of their sites.
The important thing is: they are mainly using their voting system to reach these goals by statistics and professional community control. In this sense, yes Stack Exchange is run by the community.