This is all explained on Meta.SE: New UI encourages askers to confirm or dispute duplicate votes. Here's a summary:
When there is at least one duplicate close vote on their question, the author sees a banner:
Clicking on that button (and confirming) closes the question as a duplicate.
Shog9 explains in a comment:
Although the effect is the same in this (one, narrowly-defined) scenario, question owners do not have binding close votes on their own post - I believe such a feature would likely introduce a great deal of confusion unless also coupled with binding reopen votes, which would of course introduce either an opportunity for abuse or a ton of extra rules for voting. The system is closing the question, not the user who confirmed the duplicate.