From the article-linked ruling press release - what it means in practice, what this was about:

In order to protect works covered by copyright or related rights against offences committed on the internet, a French decree introduced two personal data processing operations. The first operation consists of the collection, by rightholder organisations, of IP addresses which appear to have been used on peer-to-peer websites to commit such offences and the referral of those IP addresses to the Haute Autorité pour la diffusion des œuvres et la protection des droits sur internet (High Authority for the dissemination of works and the protection of rights on the Internet) (Hadopi) 1. The second operation, carried out by the internet access providers at Hadopi’s request, consists, inter alia, of matching the IP address with the civil identity data of its holder. Those data processing operations enable Hadopi to initiate a procedure against the persons identified, combining educational and punitive measures, which may lead to a referral to the public prosecution service in the most serious cases.

I find the ruling press release is much more understandable (and much more informative) than the OP-linked article.