The system learns to trust you based on your reputation, and gives you new abilities as your rep increases. Part of the trust relationship you develop is not just reputation, but also time: that you've used the site over a period of time, and over that period you should come to understand the community and accepted procedures for how it operates.
So the reason the reputation threshold for certain abilities is set at a particular level is because, thanks to the daily cap, the system can assume that even a well-qualified user who hits the cap every day from day zero has spent at least a certain number of days actively using the site. A particular threshold choice equates to at least certain number of days actively using the site.
For example, vote to close at 3000 means you've actively used the site on at least 15 days (accepted answers aside) — hopefully long enough to understand what closing a question really means. It gives the community a chance to assimilate the user.
It's not fool-proof. If you're Alan Kay, for example, you could post twice, come back a year later, and find you've earned quite a few privileges. But it's a pretty good measure.