I used to wonder the same thing when I was in your shoes. I used think it would be great if I could just go ahead and graduate from undergrad and move on and do my MBA. I used to hate that work experience requirement, especially since I worked in a managerial role for a majority of my undergraduate years.
When I got to b-school, however I changed my tune. Unlike Med. School or Law School, which focus heavily on didactic teaching, b-school is much more experienced based. Students teach each other more than the professor teaches the students. By having full-time experience post-undergrad, b-schooler's are able to much better determine what they want to get out of the classroom and step immediately into managerial roles at firms post-grad.
While you might feel as though you're getting some experience now as an intern (this doesn't count towards the "requirement" by the way), it's difficult to get yourself into a position of leadership, which is what b-schools really look for.
So, while it may seem annoying now, years from now when you do go for your MBA, you'll be thankful for the work experience requirement.