To understand why the CMB is partially polarized first recall the similar process of polarization by reflection off of a surface:
Polarization by Reflection
Heuristically, the incident unpolarized light "shakes" the electrons. This shaking re-radiates the outgoing reflected light. Since the electrons shake most easily in the plane of the surface, the outgoing light is mainly polarized perpendicular to the plane of the scattering. This is why polarized sunglasses reduce the glare from reflection.
Thomson scattering provides a similar means of polarizing CMB light. Consider incoming radiation from the left being scattered by 90 degrees out of the screen:
Since light cannot be polarized along its direction of motion, only one linear polarization state gets scattered.
Of course there is nothing particularly special about light coming in from the left. Consider instead light coming in from the top:
Now the outgoing radiation possesses both polarization states. If the incoming radiation from the left and top are of equal intensity, the result is no polarization in the outgoing direction.
Only if the intensity of the radiation varies at 90 degrees, i.e. the distribution has a quadrupole pattern, does a net linear polarization result:
In particular there is a net linear polarization that is aligned with the cold (red) axis of the quadrupole anisotropy.