http://arstechnica.com/tech-policy/2016/07/kickasstorrents-a...
Somewhat more/complementary details seem to be available on: https://torrentfreak.com/feds-seize-kickasstorrents-domains-... — e.g. regarding methods:
[...] The complaint further reveals that the feds posed as an advertiser, which revealed a bank account associated with the site.
It also shows that Apple handed over personal details of Vaulin after the investigator cross-referenced an IP-address used for an iTunes transaction with an IP-address that was used to login to KAT’s Facebook account. [...]
Some aspects which seem interesting to me, from what is reported:
• that apparently KAT owner tried to shield off DMCA takedown requests (which I'd see as trying to affirm being legal);
• that according to the articles he seems to not have used Tor (or fumbled in it).
(Assuming no parallel construction and that he's actually the guy, etc. etc.)
EDIT: I couldn't really find any Polish sources — suppose because it's middle of the night here... (the single article — http://www.dobreprogramy.pl/Zalozyciel-Kickass-Torrents-zatr... — seems to be written based on the above English-language ones)