I do not know how it is done except that, as Tony said, it involves digital signal processing. In my thinking about it I imagine it involves sampling of the original audio, processing to slow down the tempo (stretch out the time scale) and filling in the gaps with a version of the original sampling. In other words if you need to make a quarter note last as long as a half note (slowing the tempo by 1/2) substituting a doubled DSP version of the original note (the equivalent of playing the same note twice as long) ought to do it.

I am certain that the algorithm involved is much more complicated than my imagination. I think I had better stick to trying to play the harmonica! The sample MP3 file on the Transcribe! site is interesting to listen to. It demonstrates the sound quality of slowdown-without-pitch-change. It contains 7 seconds of a guitar solo played quite fast, followed by the same at half speed.

Michael