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What happens in transparent materials? Do their molecules oscillate with the same frequency as the EM wave and then reemit in the same direction? Or the light goes through meshes in the bulk?

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You’re basically right. In a transparent material, molecules/atoms oscillate with the same frequency as the incoming light, and they re-emit at the same frequency with little or no loss. The light beam ends up going the same direction as before because the collective action of all the emitters is like a phased array which destructively interferes any waves going the “wrong” way. There is also a slight time delay between absorbed and emitted light at each atom which gives the refractive index.

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Electromagnetic waves, light and all the frequencies above and below, are modeled by the classical electromagnetic theory of Maxwell.

In this article transparency of light is explained using the knowledge of the quantum mechanical frame of matter.

Electromagnetically induced transparency (EIT) is a coherent optical nonlinearity which renders a medium transparent within a narrow spectral range around an absorption line. Extreme dispersion is also created within this transparency "window" which leads to "slow light", described below. It is in essence a quantum interference effect that permits the propagation of light through an otherwise opaque atomic medium.

At the quantum level, light is composed by a great number of photons, ( see this interesting experiment) and the interaction of the photons with the material's molecular structure is what defines if it is transparent or not.

Hand waving, one can think of the photon scattering on the quantum molecular lattice of the material. If the material is transparent it means the photon+lattice scatter is elastic, the same frequency/energy and phase of the photon is retained, so that images can be transmitted. If the scattering Photon+lattice is inelastic the material is opaque.

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  • $\begingroup$ But why does not scattering change the direction? $\endgroup$
    – Mercury
    Commented May 12, 2022 at 5:01
  • $\begingroup$ @Mercury if you looked at the single photon at a time link you will see that the direction of the photon changes, but the quantum mechanical wavefunction solution of the problem retains the phases as seen in the classical interference pattern on the right. Photons build up light but light is the quantum mechanical ensemble of many photons. $\endgroup$
    – anna v
    Commented May 12, 2022 at 5:47
  • $\begingroup$ All good but glass is amorphus. $\endgroup$
    – Mercury
    Commented May 12, 2022 at 16:12
  • $\begingroup$ @Mercury you mean it is not a regular lattice, it still has quantum mechanical points which can enter the quantum mechanical scatter "photon +glass". $\endgroup$
    – anna v
    Commented May 12, 2022 at 18:29
  • $\begingroup$ So in the end EM photons go through the meshes in a way undergoing some elastic statteing on the way because interference happens through slits. $\endgroup$
    – Mercury
    Commented May 13, 2022 at 12:57