The way I see it is that there are two basic scenarios:
When there aren't that many users of a site who have "vote to close" rights (normally 3K+ rep, but only 500 rep on beta sites). When there are a largish number of users who can vote to close it becomes less necessary for the moderators to step in and close a question "unilaterally". Personally, I might add my "final" vote when a post is on 3 close votes, just to "speed things along".
When a question is flagged by someone who doesn't have close rights. This is an indication that someone thinks that the post is bad enough to be removed as soon as possible. The more flags there are the more "urgent" it is that the post is closed sooner rather than later.
If there are "enough" users with access to the "tools" menu (normally 10K+, 2k+ for beta sites) then these users will be able to see lists of questions with close votes and add their own vote. This reduces still further for the moderators to step in.