I do not see how this dipole anisotropy is due to the motion of the earth (and thus a doppler effect). Does anyone have a visualization of how this maps onto the earth and the motion of the earth such that I can see why it is a doppler effect?
It is said that the effect is due to the earth's motion relative to the CMB, but I have no intuition for how the CMB is moving relative to us? is this due to the movement of the solar system through the milky way?
The CMB dipole is not solely due to the Solar System's motion through the Milky Way. Indeed, the CMB dipole corresponds to motion in nearly the opposite direction, which means that the Milky Way itself is moving at over twice the speed of the Sun's orbit! Your picture is in Galactic coordinates (same coordinates as this picture). With respect to the CMB rest frame, we move toward the red spot. But with respect to the Galaxy, we move toward a point on the equator about halfway between the center and the left edge.
See Table 3 of this article (page 5). The CMB dipole tells us that the Sun moves at 370 km/s with respect to the CMB rest frame. But since the Sun moves at ~220 km/s with respect to the Galaxy in nearly the opposite direction, we infer that the Milky Way itself moves at 550 km/s with respect to the CMB rest frame.
