According to my list of privileges, I have the privilege of approving tag wiki edits. Which is correct, as my reputation is a little over 1800, while that privilege needs 1500.
While you may approve tag wiki edits at 1500 reputation, you need to attain the Trusted User level (4000 rep) to submit them without review queue approval.
Trusted users are allowed to perform trusted actions, including:
Editing tag wikis without needing approval
Thus it takes three users to get a tag edit in, one to submit the edit and two with 1500 rep to approve it. Only a user with 4000 rep can make a tag wiki edit without supervision.
It is this way because everywhere on Stack Exchange, you're given experience seeing what the qualifications are for good contributions before you're given the privilege to freely edit them.