A rolodex. It's everything. WHO ya know. And more importantly, who will write you a check to go looking for employees on their behalf.
An MBA is nice—mainly for the shmooze factor; business school is a great way to meet lots and lots of people inside of a given industry. Some headhunters dropped out of law school, or maybe even finished and worked in the laugh (read: law) field for a while—there's always demand for good lawyers who can chase ambulances, litigate global patent suits, or attract corporate paperwork dollars.