This is a simplified description that is appealing and not too wrong for high school students. It was greatly popularised by Richard Feynman in his lay book "QED: The Strange Theory of Light and Matter" as a means of introducing the idea of a medium. The more accurate description is that the electromagnetic disturbance in a medium is a quantum superposition of excited matter states and free photons, the latter still with speed $c$. If you think of the phenomenon as sequential, i.e. with the disturbance as a sequence of propagation - absorption - re-emission cycles, then it is mysterious as to how the re-emission is in the same direction as the incident light. You need to understand that the whole lot: free photon states and excited matter states are in superposition. Therefore, the emitter can in principle emit in any direction, but it has a low amplitude to do so owing to destructive interference with the other coupled states in the superposition unless the re-emission matches the incoming photon direction closely.
This is to be contrasted with fluorescence, where full molecular transitions between bound states happen and momentum, energy and angular momentum are transferred to the medium, thus the emitted light is of lower energy than the incoming light, is pretty much omnidirectional (aside from for very short fluorescence lifetimes) and partly depolarised. Angular momentum isn't transferred to the medium as readily, and thus, contrary to common belief, the polarisation of fluorescence is correlated to that of the driving light, although some depolarisation happens. This correlation is weaker the longer the fluorescence lifetime is.
I give a similar description to the above in my answer here.
In summary, although my view is often thought of pedantic by my work collegues, the locally measured speed of a photon is indeed always the universal massless particle velocity $c$. Light in a medium is not pure "light", it is a quantum superposition that, if you want to be more precise, is variously called "polariton", "plasmon" or "exciton" depending on the exact nature of the exited medium states in the superposition.