Originally Posted by Mr Mxyzptlk
Because users who view YouTube are not liable for any copyright infringement pertaining to the clip. Only the person/s responsible for the upload.
"In connection with User Submissions, you affirm, represent, and/or warrant that: (i) you own or have the necessary licenses, rights, consents, and permissions to use and authorize YouTube to use all patent, trademark, trade secret, copyright or other proprietary rights in and to any and all User Submissions to enable inclusion and use of the User Submissions in the manner contemplated by the Website and these Terms of Service"
...further to...
"However, by submitting the User Submissions to YouTube, you hereby grant YouTube a worldwide, non-exclusive, royalty-free, sublicenseable and transferable license to use, reproduce, distribute, prepare derivative works of, display, and perform the User Submissions in connection with the YouTube Website and YouTube's (and its successor's) business"
That YouTube allows the clips to be shown is exercising the above right, having stated that they are not responsible for any legal hassle pertaining to the supposed owner of the clip (ie person who uploaded it).
EDIT: Worth noting that YouTube also monitors the site for copyright infringement and removes any footage seen to do so when notified. This further lessens the liability of the viewer.