How is plane polarized light rotated by chiral compounds?
Circularly polarized light is like a helix that twists through space. The two components are mirror images of each other.
Now, every molecule interacts with both the left-handed twisting light and the right handed twisting light. The interactions differ. Every molecule, in different orientations, interact differently with the left-handed and the right-handed circularly polarized light.

Animation Reference: https://en.wikipedia.org/wiki/Circular_polarization
Now, if the molecules in solution have any mirror symmetry themselves, then if averaged over all the molecule interactions the left-handed and right-handed circularly polarized light interact in the same way, and so the polarization does not change.
But this does not hold for chiral molecules: for a particular interaction between a molecule in a certain orientation and the left-handed circularly polarized light there is no corresponding identical interaction with the right-handed circularly polarized light. There is no such molecule of that mirrored shape around.
This results in a net difference in interactions of the molecules with the left-handed and right-handed circularly polarized light, which can be described as a difference in refractive index for the two light waves. And this difference in refractive index can be detected as a change in the direction of the polarization for the sum of the two light waves.

Now, if we want to consider this using the concept of the speed of light varying in media, light goes slower in matter than in vacuum. This slowing down of light in matter is measured using the refractive index n. A higher refractive index means that the light goes slower and so keeps the light longer in the medium. Now if the medium is chiral, it gives the light two different speeds, one for the light that rotates its polarization clockwise and the other one for rotating polarization counter-clockwise. Any polarized light has only two parts (clockwise and counter-clockwise).
The two parts are combined and so the light shows a direction of polarization. When these two parts of light pass through chiral matter, one goes faster and the other slower. The result is that the polarization of light is rotated.

Response to @ Michael Seifert comment;
The way of splitting the light really depends on the matter, rather not decided by the light itself. . If matter has two directions (like a calcite crystal), light is split linearly (parallel and perpendicular). If matter is chiral, (like a solution of bio-substance) light is split circularly (clockwise and counter-clockwise). If matter is both, then all four polarizations are possible, which makes the out-going light hard to nail down.

References:
https://en.wikibooks.org/wiki/Organic_Chemistry/Chirality/Optical_activity#Why_Polarized_Light_Is_Affected
https://en.wikipedia.org/wiki/Optical_rotation
https://www.quora.com/How-do-chiral-molecules-rotate-the-plane-of-polarised-light