... Under the DMCA's safe harbor provisions, websites are protected from copyright liability for hosting videos posted by users so long as they promptly remove the infringing material when notified by the copyright holder. WHDH didn't want its weatherman creating a storm, so as the copyright owner of the news program it could request that YouTube pull the video from its site.
The DMCA's notice-and-takedown mechanism puts the onus on the individual who posted the material to send a counter-notice asking that the material be put back up and usually raising some defense to copyright infringement, such as fair use. In this case there's a fairly strong argument that the 27-second clip of Bouchard is fair. The amount of the original broadcast used is very small, the purpose of the clip is to spur public discussion, and there is arguably no effect on WHDH's news market. It's likely WHDH either didn't consider fair use before ordering the clip's takedown, or it simply didn't care.
He notes there's currently a YouTube fair-use case in federal court.