Exactly how, I don't know. But suppose they add a machine to the Bit-torrent network and then search for and find property where the IP is owned by their clients, who have (as part of the investigation) authorised Logistep to make copies. Logistep then d/l that material, and wait to track the IPs of people that d/l from them.
I'm sure it's a lot more sophisticated than that, certainly in terms of logging and audit trail, but it'll be that kind of principle, I'd expect. And, of course, no warrants required and they aren't doing anything wrong in making the copies because they have permission. People d/l'ing from them, however, don't.
In other words, it's a 21st Century high-tech form of honey-trap. You bait the trap, lay out the lure and wait for someone to bite. Then you yank hard. An in case anyone suggests it's entrapment, no, it isn't. If they actively encouraged people to d/l, if they promoted the 'service' or told people how to do it, then t=yeah, maybe, even probably, it would be. But if you lay the bait and passively wait for nibbles, it isn't.