Individual photons certainly don't have a rest frame. However, there is a rest frame in which the CMB is almost perfectly isotropic (the deviations from a perfect blackbody spectrum are of the order of 1 part in 100,000), and for convenience we call that the rest frame of the CMB.
That frame is essentially the rest frame of the plasma which emitted the CMB, i.e. the surface of last scattering, adjusted for the Hubble flow.
Our motion causes anisotropy through simple Doppler shifting: the CMB photons coming from the direction we're currently heading towards get blueshifted, the photons in the opposite direction get redshifted.
The Earth's velocity with respect to that frame is a little complicated, because we're orbiting the Sun, which is orbiting within the galaxy, which has its own motion in the local group, etc. Of course all of those motions are operating at different time scales, and different speeds. The shortest period effect is of course due to our orbit around the Sun, but our orbit speed is pretty sedate compared to the other motions I mentioned. So there's noticeable annual variation in the exact amount and location of the anisotropy, but the low period high velocity motions are the major factors controlling the anisotropy.
This famous image (from Wikipedia
shows the CMB after it's had the anisotropy corrected. The 1 in 100,000 parts variations I mentioned above are amplified enormously, otherwise the image would look totally uniform. This amplification can only be done after the anisotropy compensation, otherwise the anisotropy would totally dominate the image.