Why is glass transparent?

Photons pass through glass because they are not absorbed. And they are not absorbed because there is nothing which "absorbs" light in visual frequencies in glass. You may have heard that ultraviolet photons are absorbed by glass, so glass is not transparent for them. Exactly the same happens with X-rays for which our body is nearly transparent whilst a metal plate absorbs it. This is experimental evidence.

Any photon has certain frequency - which for visible light is related to the colour of light, whilst for lower or upper frequencies in the electromagnetic spectrum it is simply a measure of the energy transported by photon. A material's absorption spectrum (which frequencies are absorbed and how much so) depends on the structure of the material at atomic scale. Absorption may be from atoms which absorb photons (remember - electrons go to upper energetic states by absorbing photons), from molecules, or from lattices. There are important differences in these absorption possibilities:

1. Atoms absorb well-defined discrete frequencies. Usually single atoms absorb only a few frequencies - it depends on the energetic spectrum of its electrons. Regarding atomic absorption, the graph of absorption (plotted as a function of frequency of light) contains well-defined peaks for frequencies when absorption occurs, and no absorption at all between them.
2. Molecules absorb discrete frequencies but there are many more absorption lines because even a simple molecule has many more energetic levels than any atom. So molecules absorb much more light.
3. Crystalline lattices may absorb not only discrete frequencies but also continuous bands of frequencies, mainly because of discrepancies in the crystalline structure.

As glass is a non-crystalline, overcooled fluid, consisting of molecules, its absorption occurs in the 1st and 2nd ways, but because of the matter it is composed of, it absorbs outside our visible spectrum.

Essentially because of absorption. When photon flies into the material it interacts with its constituents. This interaction can be divided into two contributions. One of them is elastic and is the source of the index of refraction (because effectively it just slows the photon down) while the other one is inelastic. Photon gets absorbed by an atom (say) and later it is emitted as thermal radiation in random direction thereby losing the original information it carried.

When you look at this macroscopically, this process will be described by some parameter like penetration depth and intensity w.r.t. depth will decay exponentially. So if you made opaque objects thin enough, they would still be transparent (although the outgoing light would be weaker depending on thickness). Of course, this discussion completely avoids surface effects (reflection, refraction, scattering, etc.).

Note that all of this depends on frequency of the incident light. Atoms (let's just talk about them for simplicity; in reality there will be contribution also from molecules, lattice, free electrons and whatnot) have something called absorption spectrum. This arises because for certain frequencies electron can catch the photon and get excited to the higher energy state. So, while a material can be transparent in certain range of frequencies (like glass is for visible light) it can be quite opaque in others.

This may be a little technical, but I always thought it was cool: one of my professors once pointed out that transparency only happens because the material is (approximately) a linear dialectic over the frequency range that you care about. Turns out water is a linear dielectric over precisely the range of frequencies our eyes can detect. coincidence?