There are a couple ways to do this:
1. Manual port forwarding, it's likely that there are at least a few power users who already have the torrent and are seeding who have this set up. Seedboxes are a notable example, they're often simple servers that actually have a public IP.
2. UPNP, a protocol that lets you ask your router to set up a temporary port forward for you. Again, not all peers support this, but some do, and you can just connect to those.
3. Hole Punching. Imagine Alice is sending data to Bob, and her router ends up sending it from port 1234. Her router needs to send the packets it receives on port 1234 back to Alice's computer, to allow her to receive Bob's responses. Some routers will do this no matter which IP the packets are coming from. If Bob tells alice her router is sending from port 1234, she can spread that fact to others and let them contact her that way.
I don't know if BitTorrent clients take advantage of this specifically, but it's a very common way of doing NAT traversal in general.
NATs is why private trackers have the concept of "connectability", if you're "connectable", it means you can accept connections from other clients. Crucially, if just one of the peers is connectable, they can both communicate, so connectability is heavily encouraged but not required.