Port forwarding allows for a direct connection from one client to another. Effectively when a VPN let's you port forward if you go to the external IP address they assign + the port they assigned they will be able to directly connect to a port that a program/service is listening on.

Torrent sites are only telling your client who to connect to via the tracker, your client still needs to be able to connect to them. You can still download without this but it is generally slower (may be going through a relay) and you cannot seed torrents except for anyone that can directly connect to you on whatever VPN you are on.

*Likely some mistakes above. Keep me honest denizens of the web.