An excellent question! As materials scientists, we think quite a bit about optical and electronic properties of materials. In order to answer your question, we'll need a few bits from quantum mechanics.

You may have had some classical (i.e., Newtonian) physics already during your time in high school. Classical physics does well to describe the macroscopic world, i.e., the size of things on the order of your or me. When you get to small objects on the order of the size of an atom, you need to start considering quantum mechanical effects. One thing classical mechanics tells you is that energy exists in a continuum. For example, if you hold a ball above the ground, that ball has potential energy; you can change the height of that ball by an infinitely small amount and the potential energy will change accordingly. Now if you pretend you can do that with an electron, quantum mechanics tells you that energy is now quantized. That is, that electron can only exist at particular energy levels and only at those energy levels. This is important for understanding how light interacts with matter, and ultimately why glass is transparent.

There are three ways that light interacts with matter: absorption, transmission, and reflection. We will focus on absorption and transmission. Recall that light (or more generally electromagnetic radiation) is just another form of energy. Absorption happens when incident light has enough energy to excite an electron from its lowest energy, i.e., ground, state to a higher energy, i.e., excited, state (see diagram ).
Note from ScienceLine Moderator: The diagram has a misspelling. Instead of saying "excited states", it says "exicted states".

However, if the incoming light does not have enough energy to excite an electron to an excited state, that light will simply pass through, i.e., transmit. Emission is also possible, but for materials like glass, you are more likely to get emission via phonons (i.e., heat) rather than photons (i.e., light). In a solid like glass, these energy states form a set of energy bands and between these energy bands there may be an energy gap where no electrons are allowed to transition. For glass, this energy gap is large. What this means is that any wavelength of visible light is not sufficient to excite electrons from the ground state, and thus transmits. However, UV light is high enough energy and you would find that glass is opaque to it.

Fun fact: a similar reason why glass is transparent is also why glass is such a poor insulator!

Another fun fact: these quantized energy levels are characteristic to each element, and is in fact used to identify the composition of stars!

Visible light doesn't interact with the chemical bonds in glass that would allow it to be absorbed. This is not true of other colors of light, though: ultraviolet and infrared are absorbed by glass.