An atom where an electric field can stretch electrons off to one side a bit can become a dipole, these are called Dielectric. Energy is stored in the stretching of the position of the electrons. Atoms that are forced into a dipole are also sometimes said to be polarized. In the plane wave there is a whole volume of atoms with electrons moving up. This volume of electrons going up is like a current and produces a magnetic field around it. A half wavelength away the electrons are going down. It produces a magnetic field around it. In between the two magnetic fields add together. You can also think of it as in between it is like electrons are moving around that spot and electrons going in a circle make a magnetic field.
As the electromagnetic wave moves the electric field and magnetic field are off by 90 degrees to each other and the direction the wave is moving. They are also out of phase by 90 degrees.
Atoms with free electrons, that conduct electricity, do not propagate electromagnetic waves well.
Atoms where the electrons can move at the right frequency can conduct light of that frequency.
It makes sense that much solid matter, where electron bonds are such that the can not move so freely, do not propagate light that humans can see. We needed to see the solid objects and we don't need to see the air.