I'm not up on all the details yet, but my first impression is that this will be a massive problem for startups and small businesses.
Filtering uploads for potential copyright violations is an almost impossible technical task. YouTube might be able to do it (with a lot of false positives), but there's absolutely zero chance that a scrappy video or audio startup will be able to, which means the E.U. has just created an impenetrable barrier to starting a new online business involving user-generated content.
The U.S. doesn't have a great track record either when it comes to copyright legislation, but for all its faults, the DMCA (Digital Millennium Copyright Act), which governs how businesses in the U.S. must respond to complaints of copyright infringement, actually does a not-terrible job of balancing the need to protect copyrights while also not placing an undue burden on websites that host user-generated content.
The E.U. could have learned from the good and bad parts of the DMCA and passed legislation that improved on it, but it seems like they instead ignored the good parts and went way overboard with their approach.