It's something of a misnomer: it's not the rest frame of the radiation itself, but the rest frame in which the CMB appears the same in all directions: in most rest frames, you'll see it redshifted in one direction, and blueshifted in the other (this is what this looks like for us, for example: the overall hot/cold spots (NB: on this diagram, red is blue-shifted and blue is red-shifted, because humans like red to be hot, even though blue is hotter) are due to our velocity reshifting it, the funky lumps are local effects. If you adjust that to account for shifting the velocity of the observer, you can get it to the point at which that looks almost exactly flat (this famous image scales up the differences by orders of magnitude in comparison to the previous one - actual differences are on the order of one part in 100,000). The reference frame where that image is flattest (modulo a few adjustments for local effects) is the CMB reference frame.

An individual photon doesn't have a rest frame, but the CMB rest frame is the center-of-momentum frame of a collection of photons. Since the photons have all different momenta, their center of momentum doesn't move at the speed of light.