What is the mechanism of transparency of EM? 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?
 A: 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.
