When I visited American last spring it seemed that most CLEG kids were aiming for law school and then eventually working in the government, NGOs, or as lobbyists because the degree allows you to get an understanding of the important aspects of government (economics and the general structure/purpose of government) as well as the communications skills needed to be a successful politician all while getting a leg up on law school classes because quite a few people who enter law school didn’t major in anything directly related to law. It’s basically future politician training.
Also something important to keep in mind about the CLEG major is that it is apparently very hard to double major or even minor (depending on the student) in something else because it is such a demanding program with a rigorous course load.