Basically, YouTube as a content host, would actively, promptly and sometimes aggressively, respond to the "potential" copyright owner's requests, in exchange of keeping themselves away from lawsuit.
This way, they don't need to pre-screen the content upon the uploading, because it's up to the copyright owner to find the illegal content and report (YouTube still do that in some degree with content-ID, though).
There will be false-positives, there will be false claims, but they choose to play it safe.
Just to make it clear: this whole system is fucked up, and how "aggressive" the host needs to be is a nuance that all parties involved will have dramatically different views. I just feel like lots of people don't understand the principle of DMCA safe harbor concept. It is the consequence of current copyright law; without it, user-generated content hosts can't survive the legal trouble (or they can, but would need significant more legal resources).