That might just be pure coincidence. I guess by the time they reach 10k, those users are fully committed to the site and keep coming back often. If you also increase the volume of (good) answers that you provide every day, you can significantly increase your average rep gain, because the +15 that you get from an accepted answer is immune from the daily reputation cap.
If you have contributed a lot of answers (which you will have by the time you reach 20k), all your old answers will continue to gain upvotes from people who google the question and like your answer. This also adds to the steady stream of incoming rep. In addition, you'll find this trend to be more prominent in the higher traffic tags which have a steady influx of questions. For people active in smaller tags, you wouldn't really find a big difference in their activity & rep gain before 10k and after 10k.
There really is nothing in the moderator tools that will let you earn more reputation. Those are merely tools for doing janitorial work on the site, such as approving edits, closing posts, deleting crappy stuff, etc. We also get to review boilerplate flags in the queue, which again helps the ♦ moderators make their decision, hence helping improve the site a little. This feature certainly helps 10k+ users increase their flag-weight, but then flag-weight really is a joke anyway.