A VPN doesn't make you anonymous, it enhances your privacy. If you login to a named FB account you're still you. All you've done is tell FB you're using a VPN.
In terms of torrenting, a VPN client, if configured correctly, encrypts all data between the device its on and the endpoint. This means your ISP cannot spy on what you're doing. When you leech or seed it also masks your real IP.
But you have to trust your VPN provider isn't logging you. And if you use a provider who's located in a good country (legally speaking) but they own or rent servers in not so good countries (5 eyes etc) and you connect via those servers then the provider has no physical access to them. Tracking could easily be carried out without their or your knowledge.
If you pay a provider with a card or PayPal then there's a paper trail to you. Use a VPN that accepts Monero or cash physically mailed to them.
Make sure your VPN network interface is bound to your torrent client so if your connection drops, your IP isn't exposed.
Its trivial to find out youre using a VPN and which one and which of their servers youre using. If you pay for your VPN with identifying information (a card, PayPal etc) then they can theoretically make the provider log your specific activity.