My thought is that, on a variationally stationary path, all nearby paths are of the same length. So if you have a photon leaving a point along many possible paths, near the variationally stationary path the waveforms of the photon at each path will be lined up, interfering constructively. On the other hand, if the path is not variationally stationary, the waveform will drift out of phase on nearby paths (because they have different lengths) until the waves interfere with each other.

I don't know enough about quantum FFTs to know if there is an analogy. However, I'm taking Aaronson's class next semester so I'll be able to tell you in a few months ;)