Cosmic Rest Frame Cosmic Rest Frame is defined in the paper "Adventures in Friedmann cosmology :A detailed expansion of cosmological Friedmann equations" by R Nemiroff and B Patla as the frame at a point where average velocity of energy forms in the universe is zero. What do they mean by "velocity of energy"?
It is also written that "It is usually assumed that all forms of energy are at rest on the average with respect to all other forms of energy." How can this be justified? 
 A: EDIT: My first answer seemed to imply that radiation is at rest in the Cosmic Rest Frame. Radiation is not in rest in any frame. See below.
The sentence shouldn't be read as "[velocity of energy] forms", but "velocity of [energy forms]"$^\dagger$. The sentence refers to "energy forms", i.e. the different forms in which energy can manifest itself. These forms are e.g. dark matter, normal ("baryonic") matter, and radiation. So what the authors (and others) refer to as the Cosmic Rest Frame is the frame in which these different energy forms have no preferred direction.
A given galaxy, for instance, moves through space with some velocity, typically of the order $100$–$1000\,\mathrm{km}\,\mathrm{s}^{-1}$. But on average, galaxies have a net velocity of zero in comoving coordinates, i.e. the coordinates that expand with space. The same is true for radiation, although in this case we don't consider the photon's velocity (since that is the speed of light in all reference frames), but it average frequency. Most photons in the Universe are in the so-called cosmic microwave background radiation, the spectrum of which is the same in all directions in the Cosmic Rest Frame. Since Earth moves through the Cosmic Rest Frame at some $400\,\mathrm{km}\,\mathrm{s}^{-1}$, we actually measure a slightly blueshifted spectrum from the direction in which we're headed, and a slightly redshifted spectrum in the other direction.
As for your last question, this is an observational "fact". We can measure the velocity of the various components, and we find that, on average, they are zero.

$^\dagger$Fun fact: This confusion wouldn't rise in my language, where "energy forms" is a single word.
