I can answer your question, but if you would like details, I would have to know who your ISP (Internet service provider) is.
Tor Browser, simply put, is a repackaging of Mozilla Firefox configured for the Tor SOCKS5 proxy and maximum anonymity. When you click a magnet link, Tor Browser should warn you that an external application would open.
The external application is your BitTorrent client (Vuze, uTorrent, Transmission, etc.), which is probably not configured to use Tor. Your ISP can still see that you are downloading torrents.
When you wrote,
I have recently received a message on my computer
and
I have spoken to those who claim to be the admins
something seemed amiss to me. Did you mean that you received an email? If ISPs are to notify you that you are violating their terms of service, they ought to notify you reliably.
If you received a popup on your computer, then more than likely, a virus is tricking you into thinking that your ISP is going to terminate your Internet service.
You should know who your Internet service provider is, so you should also know how they send you notifications about their service. If you received an email, check the sender for authenticity. A fake email can be identified by lots of misspellings, a suspicious "From:" field, low quality JPEG images, mysterious links, etc.
Furthermore, I've never heard of people who work for ISPs call themselves "admins". Every time I call AT&T or Time Warner Cable, they connect me to "representatives". That's another suspicious hint.
