How do electromagnetic waves transfer energy?
Electromagnetic waves are simply coupled oscillations of electric and magnetic fields which are self sustaining and can travel through vacuum.
You must be familiar with the fact that, the energy density of an electric field is given as
A very simple way of checking this fact is to allow EM waves to pass through a surface containing static charges. The charges will be set to motion by the electromagnetic field.
In fact, H. Hertz' original EM wave detector working exactly on the same principle.