How can you slow down something without affecting pitch?

With some tricky audio coding. Same way you can change the pitch without changing the speed. It's all DSP-type stuff. You'd be surprised what you can pull off with a little clever mathematics and an audio signal.

You can only go so far before you start to hear artifacts, but for the purpose of just a little bit of pitch correction or speed change, it's practically indistinguishable from the original if the code is good. Many pro audio applications will do this, allowing you to mix two disparate pieces of music together and making them sound as if they were in the same key or the same tempo. Impressive stuff when you see it in action.

Changing an audio clip as much as "Transcribe" does (slowing it to 1/2 or 1/4 speed) introduces audible artifacts, but for the purposes of transcription, it's fine.