The specific issue with Article 13 is that it requires hosting services that allow user-uploaded content to be published to ensure that copyrighted content is not made available. The specific wording in the law is "non-availability". This is different from takedown mechanisms (that already exist) in that it requires the service to screen the content before the fact, or become liable for the content. The current situation is that the service is not liable for copyright infringement as long as they implement a takedown mechanism - that is, allow rightsholders to complain and have the content taken down. The new situation would be that the service becomes liable at the time of publication. This shift in liability is what constitutes the most significant change, and because there is no punishment for false positives (failing to publish non-infringing content) but there is a punishment for false negatives (failing to stop the publication of infringing content) this will lead to a chilling effect. If you run a service that allows users to upload any kind of content, you would become liable for their copyright infringements.

The talk about filters comes directly from Voss's (the law's author's) blog, where he explains the law and what is intended with it. There is a specific reference to google's content id as an example of the kind of measures services are expected to implement. In the law text these are refered to as "effective technologies". The specifics of how the filter is implemented is not exactly relevant in this case, but another aspect is - currently nobody except google has an implementation of this technology, and google's implementation has very bad recognition rates and is also easily circumvented. So if you run a service that allows user-uploaded content, you now either pre-screen each post manually, license a buggy filter from google, take the risk of being sued for something your users posted, or block user-submitted content entirely. The last one is the only option for things like small forums.

Reading specifically into the law's author's blog post about this, they state that if something is wrongly filtered, the recourse is attempting to force the service to accept the upload by taking them to court. Clearly this is not something you can do on a routine basis, so if your content gets wrongly identified as infringing (and of course automated systems cannot recognize parody, or remixes, or similar but unrelated content as non-infringing) you'll end up effectively prevented from posting it.

So yes, you're right that there are already mechanisms to take down infringing material. The added change in this law is that it makes publishing infringing material the point of liability, and so each service becomes responsible for all the content that its users post. This is the significant change.