Yes, you can download YouTube videos using hundreds of sites, scripts, extensions and applications and that's because YouTube's interface didn't offer a download option. Initially, the only format available for download was FLV, which was only intended to be played using Adobe Flash. Then YouTube started to test higher-quality versions for videos, including MP4 (H264 with AAC audio). Now you can export all YouTube videos as MP4 and it would be trivial for YouTube to add this option, which is already available at Google Video.
But things are starting to change: the videos from Barrack Obama's YouTube channel ChangeDotGov have a small download link below the player. "YouTube is rolling this out slowly, initially with content that aspires to be consistent with principles of open government. I'm told it will be offered more generally. In any case, it is an important development. There have always been hacks for slurping down YouTube videos. But it is a valuable step that YouTube encourages and supports this sharing," notes Lawrence Lessig, Professor of Law at Stanford University.
Why would you download videos? To be able to watch them without having an Internet connection, to download them faster using download accelerators like Flashget or Free Download Manager, to use some of the content in your videos or to have an offline version just in case the video is no longer available at YouTube.