So I was curious to find out how Hamachi circumvents NAT (most of the time) when establishing a tunnel between peers.
When establishing tunnels between the peers, Hamachi uses a server-assisted NAT traversal technique, similar to UDP hole punching. Detailed information on how it works has not been made public. The vendor claims "...to successfully mediate P2P connections in roughly 95% of all cases ..." This process does not work on certain combinations of NAT devices, requiring the user to explicitly set up a port forward. Additionally 1.0 series of client software are capable of relaying traffic through vendor-maintained 'relay servers'.
Edit: If someone can provide a better explanation of this, please do. It isn't the best.