The rest is the "safe harbor" provision of the DMCA. Effectively, sites aren't liable for user generated content if they respond to official DMCA takedown requests in a timely manner. YouTube also goes beyond that to directly work with copyright holders to preemptively remove infringing content with content ID, which scans everything for violations, and their own tools to report infringement. They don't need to do that for the DMCA protection, but it's probably cheaper at their obscenely large scale.