They stopped touring in 1966, and it was because they were tired of the commotion at the concerts mixed with the backlash after lennon's jesus remark and the drama with the royal family in the Philippines. Lennon wanted to stop primarily because of all the screaming at the concerts no one could hear the music anyway, not even them while playing it. So, they decided to concentrate on recording.
This is all true but in subsequent years they gave as one of their reasons for not performing live that the music would be impossible to faithfully reproduce on stage. The original idea of LET IT BE was to do an album that could be done live, and give a huge concert. They couldn't quite get it together so they just performed on the rooftop (as can be seen in the documentary).
In any case, the Fab Faux is awesome.