The instrument is actually the Optigan which is similar to the Mellotron in that it played back pre-recorded loops. Instead of using tape loops like the Mellotron, the Optigan used Optical disc that you inserted into the unit like a record. You would purchase discs which were sold in different musical genres though I do not recall this specific disc title.
If you can't find a real Optigan, G-Force sells an Optigan sample library that plays on thier M-tron VST instrument.
Mark Linkous was a big fan of the Optigan and I assume he had something to do with it being used on this tune?