What is occuring on the quantum level when a molecule rotates plane polarized light?
As mentioned in the comments, the answer to your $2^{nd}$ question regarding why optical rotation isn't simply cancelled out in solution due to random orientations is given in ManishEarth's answer to Molecular chirality and optical rotation

For your first question, I will try to explain what actually causes rotation when light impinges on an optically active molecule. For chiral molecules, it is simplifies the picture to think of each enantiomer as a screw of negative charge that is either right or left handed. Plane or linearly polarized light can always be decomposed into circularly polarized clockwise and counterclockwise components. When the light collides with the molecule, the clockwise component and counterclockwise components interact differently with the electric field of the molecule, leading one of these components to travel faster than the other, which alters the angle of the plane polarized light. A good visualization of this effect (for a whole solution rather than an individual molecule) can be found at http://cddemo.szialab.org/. 

As an interesting aside, you might imagine that because we have a reasonable understanding of why optical rotation occurs, we should be able to predict, either heuristically or from a first principles calculation, the magnitude and direction of optical rotation for a molecule just based on its structure. This, thus far has not been the case. Not only are there very few ways of heuristically guessing the magnitude/direction, but quantum mechanical calculations at high levels of theory have proven unable to even consistently obtain the correct direction of the optical rotation. Given a particular chiral molecule, we still can't predict which direction, let alone magnitude, that a particular enantiomer will rotate light. 

My main point in this little tangent is that even something as seemingly simple and in such widespread use as optical rotation is still not very well understood and is a difficult area of active research.