When you look at the Sun in white (visible) light, what you see is actually sunlight that has scattered from free electrons in the solar corona. This is known as "Thomson scattering".
If we view the corona around the Sun (from our perspective), then the scattered light that reaches us has scattered through a large angle (roughly 90 degrees). The sunlight incident upon the electrons in the corona is unpolarised (that is, the electric field vector of the light is randomly oriented but at right angles to the direction of travel of the light). Such light accelerates the free electrons which re-emit light (this is what scattering is). But the scattered light is polarised when it is scattered through a large angle, because we effectively can only "see" the electrons accelerating along a line that forms a tangent to the limb of the Sun. The component of the acceleration along the line of sight does not cause any radiation to be emitted towards us.
The light we see from the daylight sky is also produced by scattering. In this case it is "Rayleigh scattering" from electrons that are bound into molecules and atoms in the Earth's atmosphere. The polarisation effects of this scattering are almost identical in nature to that caused by Thomson scattering. However, when we look at the sky right next to the Sun, then this will have been scattered through a very small angle. In these circumstances the scattered light will be unpolarised.
Thus we have a means of seeing a contrast between (polarised) light coming from the corona and (unpolarised) light coming from the sky in the same line of sight. By viewing the coronal area using a linear polariser, and then presumably rotating the plane of polarisation and taking a series of pictures, and comparing this with images taken without a polarising filter, we can isolate the small fraction of light from around the Sun that is due to Thomson scattering in its corona.
You made the comment that you don't see the sky darken when looking through a polarising filter. Well, either your filter isn't working or you were looking at the sky roughly 90 degrees away from the Sun and had the polarising filter in just the right orientation, or your sky is very polluted (the scattering caused by particulates does not cause the same degree of polarisation). Daytime sky is most definitely polarised depending on angular separation from the Sun. This map shows how the degree of linear polarisation changes with position as the Sun moves across the sky. The Sun is where the black circle is (black indicating unpolarised light). See also this answer.