The song that Thom and Tommy play here is called Cannonball Rag and it's a well known song that was actually recorded by Thoms father Merle Travis back in the 1940s.
Cannonball Rag have come to be known as the thumbpickers anthem by some and many believe that it was Travis that made Cannonball Rag. Now Merle Travis didn't compose the song Cannonball Rag. Kennedy Jones or Jonesey composed Cannonball Rag but he didn't record the song. Travis recorded the tune in the 1940s, and because of that he received the credit that way.
Anyhow Thom and Tommy play this Cannonball Rag very fast and this is not a fingerstyle song that you should try and play if you are new to guitar. It will take some time on the guitar to be able to play Cannonball Rag at this speed. But if you want to learn there are some good guitar tabs around that will help you out.