Instead of finding a pattern of chord changes and singing a melody over it, Simon and his longtime engineer Roy Hallee traveled to South Africa and met local musicians there. Simon would jam with the musicians and Hallee had set up the studio they rented so that individual instruments were recorded to individual tracks.
Later, they returned to the United States and could manipulate the jams into compositions, adding an instrument or harmonic line here, and taking it out from somewhere else. This is why the music is not that hard to learn instrumentally: the same chord changes continue throughout songs like "You Can Call Me Al" or "Under African Skies" -- although the arrangements change to signal a chorus or verse.
This track by track recording process also explains why the album sounds simultaneously like an exuberant jam session and highly produced. Listen carefully and you'll hear lots of backwards guitar lines (they start long and quiet and then get loud before cutting off sharply):
Listen at 0:32 of "Under African Skies":
And in case you were wondering, it is possible to play this line live:
So how did Simon think to create his album in this fashion? Well, we know he was inspired by a cassette tape of Ladysmith Black Mambazo, his collaborators on a number of tracks. According to Marc Eliot's Paul Simon: A Life, that cassette joined the Talking Heads' Remain in Light and David Byrne and Brian Eno's My Life in the Bush of Ghosts in his regular rotation (p. 186-7). Eliot points out that those two albums share some of the African rhythms and arrangements that would influence Graceland, what he fails to mention is that Eno and Byrne had used the same jam session and tape splicing technique that Simon would use to create their tracks.
I would not have thought to put the Talking Heads in a category with Graceland, but it almost, sort of makes sense. Simon's stream of consciousness lyrics are even fairly similar to Byrne's -- and a departure from the highly constructed poetry that he strived for earlier in his career.
You know what happens when you Google "Paul Simon David Byrne"? You find some videos of encores from Simon's June 6, 2011 show in NYC's Webster Hall. Byrne with Simon's band doing Byrne's "Road to Nowhere":
Bonus: This page has videos of Byrne doing "Al" and "I Know What I Know" with his own band. For more of a downer, that page also has an interview with Los Lobos' Steve Berlin about "collaborating" with Simon; if you want to have warm glowing feelings about Simon, you'd better skip it.