How does a single voltage reading from a microphone turn into a whole audio spectrum?
When you pause an audio file in an editor, the spectrum display is that of the audio surrounding that point in time, not the spectrum of one instantaneous sample (because as above, one sample is not enough information to contain harmonics).  Of course this varies from one program to another, and only the heavy pro programs come with tech support that will tell you how their software actually "thinks".  But in roughly the same way your ears and mind hold a snapshot of a complex sound when you hit the pause button, the software does sorta the same thing.