If you're curious about whether using clips from the debate to create the video was legal, the answer---as usual---is "it depends." The Gregory Brothers may have licensed the video, or they may have decided to rely on fair use as grounds for why they didn't need to ask permission in the first place.
For more on music and video licensing issues, check out these books in the law library:
- Music Licensing Rights and Royalty Issues
- Getting Permission: How to License & Clear Copyrighted Materials Online & Off (e-book!)
- Hey, That's My Music!: Music Supervision, Licensing, and Content Acquisition