An unpolarized wave can be represented by the sum of two waves with perpendicular polarisations and equal amplitude. The unpolarised incident wave can be considered to be of this nature, with one polarisation in the plane of incidence (p-polarised) and another at right angles to that (s-polarised), and with the electric fields of both being perpendicular to the direction of incident wave motion.
When the electric field of the incident wave is incident upon the interface, it sets up an electric field in the medium. Because of the continuity conditions for the electric field (any components tangential to the surface are continuous) and the requirement for a fixed phase relationship between the incident, reflected and transmitted waves, we obtain the law of reflection and Snell's law of refraction.
One way to think about how the fields of the reflected and transmitted waves are produced, is to imagine that oscillations are set up in small electric dipoles in the medium. The oscillations are driven by the electric field in the medium, i.e. the electric fields of the transmitted wave. These oscillating dipoles then re-radiate electromagnetic waves. Since the incident wave consists of both p- and s-polarised waves then the dipole oscillations will also have components in the plane of incidence and at right angles to that, but must also be perpendicular to the direction of the transmitted wave.
The key property of electric dipole radiation here is that no radiation is emitted along the oscillation axis of the dipole. This means that, at the Brewster angle, the electric field of the reflected travelling wave contains no contribution from the dipoles oscillating in the same direction as the reflected wave is travelling, but only from the dipole oscillations at right angles to that. As a result the p-polarised incident light is not reflected at all and the reflected wave is solely s-polarised.
At other angles the dipole oscillations in the medium are not exactly aligned with the reflected wave direction and so the p-polarised light is reflected to some extent; resulting in a partially polarised reflected wave. Note that the s-polarisation is not affected by these considerations since the dipole oscillations due to this polarisation are always at right angles to the reflected wave direction. Note also that because the oscillating dipoles would in general have different amplitudes in each direction, then the transmitted wave is always partially polarised for light incident at the Brewster angle (see https://physics.stackexchange.com/a/294528/43351 ).