Why is light subject to the doppler effect? Unlike sound propagating as motion of another substance, my understanding is that photons carry the electromagnetic energy themselves. And if I'm not mistaken, each individual photon "has" a specific frequency.
So when a photon source is moving away from an observer, why does redshift occur? Wouldn't the photons themselves (each carrying the original frequency) just arrive less frequently?
 A: If you think of a photon as a wave then the Doppler shift is what it is for any wave: an observer traveling away from the source (or equivalently and inversely,  a source going away from an observer) will see the light redshifted because it is also a wave and the observer sees fewer cycles in the same period of time. 
So you're really asking why is a photon, or any particle, also a wave, i.e., why is the Complementarity principle, that particles and waves are different ways of describing the real thing, which in all cases is a quantum field. For light the quantum field theory is QED, quantum Electrodynmaics, and in the Stack Exchange link What is the relation between electromagnetic wave and photon? it is explained how the two descriptions off complementatry or dual ways of describing the way quantum fields behave. In essence you can sometimes observe their wave-like properties, and other times their particle-like properties. See the Wikipedia Complementarity principle in Wikipedia at https://en.m.wikipedia.org/wiki/Complementarity_(physics)
Since it turns out to be true for any wave, it is not due to the kind of wave it is, it is simply that time is passing slower for a moving observer, his clock is going slower with respect to a fixed observer. How does he explain that if he's thinking of photons? Well, the same way Einstein did, with Special Relativity. 
How do you measure a photons wavelength or freq? Not with a ruler, you let it interact with another photon and see their (wave-like) interference pattern, or you left it get absorbed by some material and measure it's (particle-like energy and momentum and calculate it's frequency from it. 
The wiki article on light has a nice description and figure for the Mark-Zehnder Interferometer, which can be changed slightly to measure either the energy or wavelength, and shows that a photon is both particle -like and wave-like. See it at https://en.m.wikipedia.org/wiki/Photon, in the section on Complementarity and duality. 
