How does light travel through glass?

The first thing is that any solid object that looks solid to us is actually has huge amount of space in it...

Even in an atom, the nucleus of the atom is about a hundred thousandth of the size of the actual atom. So there's immense amounts of empty space only containing electrons, which are even smaller than the nuclei, so there's lots and lots of space for things to travel through, as long as it doesn't interact with the nuclei or electrons.

A light wave is actually quite big compared to the size of an atom. It's a quantum mechanical object - it's kind of a particle, but it's kind of a wave. You can think of it as wave which only arrives in particles - not really something with which we have a handle on.

It's a lot easier to think of it as a wave in the circumstance. The only way to stop a wave is with something which will actually absorb it or scatter it, and in something like glass there's just nothing there which will absorb or to scatter it. So it just carries on going in a straight line.

Addendum for clarity:

Light is an electromagntic wave; that means that it comprises a changing electric field, which produces a changing magnetic field, which produces a changing electric field and so on... This is how the light propagates through space. 

When light rays interact with an entity, like a piece of glass, the electromagnetic wave causes the electron clouds in the material to vibrate; as the electron clouds vibrate, they regenerate the wave. This happens in a succession of "ripples" as the light passes through the object. Because this process takes time, that's why light slows down slightly in optically more dense materials like glass. 

Different colours of light have different frequencies; light that is visible to us passes through glass because the arrangement of the atoms in glass means that they can sustain the ripple effect described above at those frequencies, so light can pass through.

But other materals, with a different configuration of atoms in the crystal structure, cannot permit light to propagate and instead absorb the energy. A good example of this is an X-ray. It will go straight through a human body mostly unchanged, but the lead apron worn by the radiographer stops it in its tracks.