I'm missing something. Here's my reasoning so far, which has left me in a hole:
The doppler effect is observed with light waves, changing the observed frequency. The energy in a single particle of light equals h times frequency. So, if the observer is moving away from the light source, then the energy they receive from a single particle of light will be less than was sent. Or more if they're moving toward it.
This, of course, doesn't make sense. Please explain what and where I've screwed up.