People breach copyright laws by uploading videos to YouTube in much the same way as they illegally use P2P software such as Limewire. That's because they don't care about copyright laws and/or because they assume that nobody will do anything if they break the law.
YouTube faces legal action from the owners of copyright material if it is seen to allow unauthorised materials to appear on its site. Some major copyright holders (such as the BBC) have negotiated to receive a licence fee from YouTube, so their videos can stay online. However, YouTube has to act very quickly to remove any videos which are derived from Paramount or Dreamworks movies, or from MTV or VH1 programmes because those brands are owned by Viacom which constantly seeks to take legal action against YouTube for breaching their intellectual property rights. (There are many other companies, including the recording companies which hold the rights to many music videos, who will sue YouTube if their videos appear online).
Chris