Why do only chiral molecules rotate the plane of polarized light and how do they rotate it?
See also: How do Optically Active Compounds Rotate Plane Polarized Light?

This is because optical rotation is a chiral phenomenon. 

Take a molecule, and draw arrows depicting the polarization of incoming and outgoing light on it. Even if the molecule is achiral, the molecule with the arrows is chiral.

Chirality can't spring out of nowhere, it can only arise from a chiral process.

Also, see my answer here. Since molecules will exist in all rotations in a given fluid, for an achiral molecule the mirror image molecules cancel each other out.