How exactly do capacitors work?
The electrons don't actually pass through the capacitor. As one plate of a capacitor gains electrons, that creates an electric field that repels the electrons of the other plate, and it's those electrons that go on to move through the stuff on the other side of the capacitor.

You're right in saying that what actually propagates is an electromagnetic field, not the electrons themselves--the electrons do move, but the EM wave moves significantly faster, and if you had to wait on electrons moving it'd be faster to send your message by snail mail than email! The key thing is that the electric field exists outside of conductors whereas mobile electrons (usually!) do not, so an electric field can couple the two sides of the capacitor together.