The reason that light travels more slowly in a dielectric is because it interacts with the electrons in that dielectric.
Light has an oscillating electric field, and if any charged particle is in the path of the light that particle will feel an oscillating force due to the oscillating electric field of the light. The result is that the wavefunction of the light becomes mixed up with the wavefunction of the charged particle.
In a dielectric the light interacts mainly with the electrons because (a) they are lighter and more mobile than the nuclei and (b) there are generally a lot more of them. So we are dealing with the interaction of the light with electrons. The result of the interaction is that the light is no longer just light. It is an entangled state of light plus electrons. This entangled state no longer has a zero mass so it travels at less than the speed of light.
The more strongly the light interacts the greater the effective mass and in cases where the interaction is very strong the light can be brought to a complete halt. This can be seen when light interacts with Bose-Einstein condensates, and indeed made the headlines a few years back when experimenters actually managed to bring light to a complete halt.
When the interaction is strong the light plus electrons forms new quasiparticles called polaritons, though for weakly interacting systems like light in glass the quaiparticle description isn't very useful.