Frank summed it up. Basically, it sounds like you were using a P2P network, a torrent. The problem is not the downloading, but that you're also uploading at the same time. Sharing what you're downloading. THAT'S the copyright infringement. How did they find out? Everyone that shares a torrent shows their IP address in the torrent tracker. It's easy enough for them to join the torrent and look at the tracker to see who all's in the sharing. Now, the IP address, itself, only goes so far to show what ISP (provider) is being used. But the ISP shares liability for simply being the conduit. So they willingly comply with the copyright holder by passing on the holder's complaint to remove their own liability.
The answer is easy. As soon as you get notice of it, immediately cut that torrent. Whether or not it's complete. And forget the honor among thieves about giving back what you've received. It's YOUR butt that's on the line. Remove all those files from your computer. Also remove any tracks of your activity, such as the torrents, themselves. They'll also hide in the depths of unknown folders, so use a free resource like CC Cleaner to help you clear out those tracks. Then wait. Most likely, once you've disappeared from the torrent trackers, they lose interest because there's plenty of others to chase after.
Don't offer to pay. Don't even offer to consider it. Don't even go to any site to read their offers. You might as well paint a sign on yourself, "Yeah, I'm guilty, just stick it to me!" Basically, things can happen by mistake or misunderstanding. They are required to give you the chance to correct. Hence, the Cease and Desist order. You've corrected it, not much left for them to chase. You have to repeatedly ignore them to show that you're too stupid to take the advise. Of course, they'd like it even more if you just rolled over and gave them money right off the top. But make them work for it. Once you get an actual summons for court, then you can negotiate. They'd STILL rather get money the easy way than go through a lot of time and expense in court. MOST likely, they wont' bother.
I've probably gotten near a dozen of such notices over various times in the last couple years. Nothing's happened. But do avoid downloading for a couple months before trying it again.