In Privileges, it says I need 2000 Reputation points to be able to edit questions and answers.
However, at my current level below 2000 points, I am already able to do this. Or, is my understanding wrong?
Are you sure that you are able to make edits that don't need prior approval before they are applied?
Every user can make 'suggested edits' but if you have >2000 rep you are don't need to get the approval of the community first. In fact you should then be able to review suggested edits yourself.
You can edit questions and answers, but they go to a queue, from which moderators and users with enough reputation (over 2000) can accept, discard or improve your edits.
Reviewing suggested edits
In addition, users with this privilege level also get access to the suggested edits queue. The queue has a list of posts which have been edited upon by users who don't have edit privileges. This queue can be accessed over at the suggested edits tab in the review section. Users with this privilege level get access to the Low Quality Posts list where they can improved the low quality posts, or suggest they are deleted.