You definitely do not need a degree, but it does help to be a lawyer. About half of the lobbyists I meet are lawyers, but the bigger predictor of success is how personable you are.
I got my undergraduate degree at the University of Minnesota. I majored in Philosophy and Biology. I went on to Vermont Law School to get my J.D. and while there I interned with the company I am currently at. After I passed the bar, my company offered me an initial position and several years later I worked my way up to the Executive Director role. I work on behalf of an association, but I am the primary lobbyist for the organization.
Lobbying typically involves going up to the state capitol and meeting with legislators. The goal is to educate policy makers about your issue and to persuade them that what your client is doing is in the best interests of the state and that individual's constituents.
Having the ability to see your work result in actual policy that results in actual benefits to the world is very enriching.
Sometimes it can be a frustrating job because there are a lot of factors outside of your control. You can do perfect work, but the bill may fizzle for all sorts of reasons, and sometimes those reasons may not even have to do with the bill you are working on at all.