Not all glass is transparent, since some of it is colored, meaning that it absorbs some frequencies of light. Nevertheless, as you say, glass that doesn't absorb light does transmit it. In that way its similar to transparent crystals, such as diamonds.
The reason is basically that the index of refraction of the glass is very nearly uniform on distances as large as the wavelength of light. That means that the light waves transmit smoothly, not bouncing off different directions. Although the individual atoms in the glass would scatter the light in different directions, just as a stick in water will scatter a water wave. However if you put a lot of regularly placed sticks in water, much closer than the wavelength of the water wave, the scattered waves from the different sticks will not be in phase except in the forward and backward directions. That means that when you add the waves in other directions the crests and troughs will cancel.
Something very similar happens for light hitting glass. Some of the light bounces back from the surfaces and some transmits through. It doesn't scatter off to the sides, except for a small amount due to small unevenness density of the glass.
(published on 05/31/2012)