It's pretty tricky and interesting. I'm sure I have some details wrong, but the overview is this:
The programs can already talk to each other through Joel's server, so they can exchange information with each other and Joel's server. Further, Joel has their external IP addresses, and they give joel information about their internal IP addresses.
They decide to try this hole punch technique. Computer A initiates a TCP connection with Computer B using B's external IP address. It won't go through, but what it does is tell's A's router that it needs to allow incoming packets from B on a given port.
Computer B does the same thing, but its message gets through to A since A's router opened a port/ip combination that matches what B sent (there's some port magic that happens here - this is non trivial, but doable).
B's router remembers that B initiated a connection with A on a given port and IP, and so A's packets now flow into B past their router correctly as well.
So it's actually pretty straight forward, but the implementation has details, especially regarding how ports are given to new TCP connections, and how NAT routers typically deal with TCP requests and how they map to external ports. These details are the interesting, and difficult, bit.
-Adam
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