RichardR's link, which mentions "heat" and "interference" of the crystal formation, implies that the effect is enthalpic, i.e. to do with interruption of intermolecular forces. My understanding of it is that one observes a lowering of the melting point even in ideal solutions, where the enthalpy of mixing is zero.
There is an entropic explanation. Consider pure water. Lots and lots of H20 molecules, as far as the eye can see. The melting/freezing point of water represents the tendency to go from a state of low entropy to higher entropy, which we can identify with an increase in disorder. As the solid will consist of H20 molecules locked into position (apart from a bit of vibrating), an increase in entropy will occur if the ice melts and forms a liquid, where there is some temporary short-term order, but in general, the water molecules are free to jiggle about almost as much as they please (albeit not as much as in the gas phase).
Now we add our salt molecules. In the liquid phase, not only can the water molecules jiggle about with gay abandon, the presence of the Na+ and Cl- ions adds an even greater amount of disorder to the solution. Of course, it also adds to the amount of disorder in the solid phase, but the increase in disorder is greater for the solution than the solid.
Hence, upon adding salt to pure water, we increase the tendency for the solid to become a liquid. This manifests itself in a lowering of the melting point. Once you have a bit of salt in contact with the snow, it will cause it to melt, which will dissolve the salt and spread the brine further, continuing the melting.
As to the question of why use salt in particular, I don't know. I assume it is to do with its cheapness, abundance, solubility in water, and also that it reduces the melting point by a useful amount.