My reading of the article is that providers are "made aware" by the rights holders, not by general monitoring of communications on their network.
It’s true that Article 15(1) of Directive 2000/31 prohibits the imposition of an obligation on an ISP to carry out general monitoring of information that it transmits on its own network.
Sounds to me that, in practice, rights holders will notify providers of suspected infringement, triggering their requirement to report to authorities, and it goes from there.
I'm not sure how this would work for a VPN provider. It seems that rights holders could only notify them of suspected piracy websites, as client traffic would be invisible to them. I also wonder how Italy can enforce their laws on the providers outside their jurisdiction, beyond compelling IP blocking to all non-compliant VPN servers in the world.
I have only performed a cursory, sleepy reading of the article, and I didn't follow the links to the relevant legislation. Happy to be corrected.