Before samplers: music made with razor blades and scotch tape

The same year this mix was made, Ultimate Soundtracker was released: http://www.youtube.com/watch?v=5ywaCR5Tg4A

Commercial sampling keyboards were available for the big budget productions around the late 70s(the big breakthrough being the polyphonic Fairlight CMI in '79), but memory prices dropped low enough to make mass-market products like the Casio SK-1 available by the mid-80s. So my impression(not having been around for that era) is that extensive tape mixes like the one in the video were primarily inspired by digital samplers; since at that time, samplers couldn't hold entire songs, tape splicing technique improved to allow the same concept to be used in a longer format. The move to computer-centric editing came with multitrack editing of entire songs(e.g. Pro Tools) which happened during the 1990s.

This video gives you some idea of how this could be done: http://www.youtube.com/watch?v=3Z7jwZK2-Gw however most people doing this kind of analog audio editing with tape actually had to cut the tape by hand, and all had to make careful calculations to keep the edits in musical time.