The one you hear an excerpt from and the one to buy is the recently released first volume of what will be a 3-volume set from the University of California. Here is the link to the Kindle version of
The Autobiogrpahy of Mark Twain.
This is not in the public domain. Twain's will prohibited publication of the unexpurgated version until 100 years after his death. Other versions of his autobiography are expurgated versions.
For an excellent review of the first volume (which is all that has been released so far), see "
His Own Best Straight Man" by Andrew Delbanco in
The New York Review of Books.