How does transparency work?

Why materials are transparent to some wavelengths and opaque to other? How does light pass through something anyway?

This takes us back to the question: Where does the coupling constant "e" come from?
Why is it always approx. -0.08542455 (or if you prefer, the inverse of its square, approx. 137.03597)?
It has been a mystery  ever since it was first discoveres more than 70 years ago.