Understanding the CMB background as a reference frame
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 long 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.
First of all, how is it possible to think CMB, a bath of blackbody (BB) photons in space, as a reference frame? Doesn't quite match my Newtonian notion of reference frames.
But it does. The references frame of $X$ is one where $X$ has zero momentum, and a photon bath has a measurable momentum and by choosing the right frame that can be made to be zero.
What do we mean by anisotropies in a BB spectrum? Tiny departures from BB distribution?
and
Why should our motion cause anisotropy?
Isotropy is a condition of being the same in every direction. From the point of view of observers not in the frame of the CMB, photons from one direction are blue shifted, those from the opposite direction are red shift1, and those not on the line of relative motion are partially Doppler shifted and exhibit angular aberration (search term "Lorentz focusing").
All of these effect make the view different in different directions.
1 Of course different wavelength means different momentum, so these effects are linked to being in frame where the CMB has a different net momentum.
If you are in motion with respect to the CMB, you'll see the photons in front of you blueshifted a bit, and the ones behind you redshifted. We do see this, and the Earth appears to be moving ~600 km/s with respect to the CMB.