Have CMB photons "cooled" or been "stretched"? Introductory texts and popular accounts of why we see the "once hot" CMB as microwaves nearly always say something about the photons "cooling" since the Big Bang. But isn't that misleading? Don't those photons have long ("cool") wavelengths because space expanded since they were emitted. There's no separate "cooling" process, is there?
 A: Cooling and stretching essentially mean the same thing here. The temperature of any blackbody radiation is related to the peak wavelength by Wien's Law $$\lambda_{\mathrm{max}} = \frac{b}{T} $$ Therefore as the universe expands, all of the photon wavelengths get stretched out and so does the peak wavelength. $$\lambda \propto a(t)$$ This decreases the temperature of the radiation by the same factor that expands the wavelength of the photons. $$ T \propto \frac{1}{a(t)} $$ This factor $a(t)$ is the scale factor of the universe and increases with time for an expanding universe.
A: I think the best way to think about it is that the sentence "the photons have cooled" is simply describing a fact, not explaining that fact. At early times, the photons at any given location had a thermal (blackbody) distribution corresponding to a high temperature (as measured by observers at rest in the natural, comoving reference frame). At later times, the photons at any given location had a thermal distribution corresponding to a lower temperature. That's what we mean when we say that they "cooled."
Of course, it's then very natural to ask why they cooled. That's where the "stretching of space" explanation comes in. I think that that explanation is problematic, as I wrote at great length here, but others disagree.
A: I will use most of @Ted answer to describe 'hot' but I will ask a more basic question:  

I think the best way to think about it is that the sentence "the
  photons have cooled" is simply describing a fact, not explaining that
  fact. At early times, the photons at any given location had a thermal
  (blackbody) distribution corresponding to a high temperature (as
  measured by observers at rest in the natural, comoving reference
  frame). At later times, the photons at any given location had a
  thermal distribution corresponding to a lower temperature. That's what
  we mean when we say that they "cooled.
  Of course, it's then very natural to ask 

Q : What would happen if CMB photons in the past were really hot?
A : It is an impossibility!
as seen here: Remaining Problems in Interpretation of the Cosmic Microwave Background :  

the expected CMB temperature increases would be prohibitive to star
  formation in galaxies at redshifts higher than $z=2$ where nevertheless the
  cosmologically most relevant supernovae have been observed

The official interpretation of the CMB, in a hot past universe, is in trouble. One good reason to consider that larger atoms in the past radiated at larger wavelengths, explaining the cosmological red-shift as is demonstrated here: A self-similar model of the Universe unveils the nature of dark energy 
