What is the mechanism behind the slowdown of light/photons in a transparent medium?

The easiest way to get the exact behavior is from thinking about light as a classical wave interacting with the atoms in the solid material. As long as you're far away from any of the resonant frequencies of the relevant atoms, this picture isn't too bad.

You can think each of the atoms as being like a little dipole, consisting of some positive and some negative charge that is driven back and forth by the off-resonant light field. Being an assemblage of charges that are accelerating due to the driving field, these dipoles will radiate, producing waves at the same frequency as the driving field, but slightly out of phase with it (because a dipole being driven at a frequency other than its resonance frequency will be slightly out of phase with the driving field). The total light field in the material will be the sum of the driving light field and the field produced by the oscillating dipoles. If you go through a little bit of math, you find that this gives you a beam in the same direction as the original beam-- the waves going out to the sides will mostly interfere destructively with each other-- with the same frequency but with a slight delay compared to the driving field. This delay registers as a slowing of the speed of the wave passing through the medium. The exact amount of the delay depends on the particulars of the material, such as the exact resonant frequencies of the atoms in question.

As long as you're not too close to one of the resonant frequencies, this gives you a really good approximation of the effect (and "too close" here is a pretty narrow range). It works well enough that most people who deal with this stuff stay with this kind of picture, rather than talking in terms of photons. The basic idea of treating the atoms like little dipoles is a variant of "Huygens's Principle," by the way, which is a general technique for thinking about how waves behave.

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Here is a nice explanation.