Will they be enforced to check every upload, or just comply to legitimate removal requests?
There's a few parts -
1. Claims - Under Art 13, if YT get a legitimate removal they must act or be fined - this is what is currently in effect anyway. If that previously "taken down" video reappears then the EU can (but may not) fine YT for allowing the material to keep appearing.
2. Enforcement - There is no enforcement information in the Article 13 so far (well technically it's now called Article 17 -
you can read it here from page 121 but it's very "legalise" - 700kb PDF), in the same way that there is no global/EU "copyright database" for companies like YT etc to check against at the moment, so as it stands right this minute it's done via "claims" just the current YT system.
So far it sounds just like the system YT already use, right? Yeap it is, only now instead of the uploader getting in trouble (bans/court/fines etc) YT themselves are now liable by EU law.
So for every single copyright "claim" the EU can (if they choose) fine YT for every single breech thats currently reported (maybe automated in the future, who knows). So what YT have have warned is that, because it's impossible for every single video, past, present and future to be checked for every single form of copyrighted material (image/text/sound etc), it would just be easier/cheaper for them as a company to ban/restrict all EU accounts rather than risk the fines.