How do electromagnetic waves emit energy?

Electromagnetic waves such as visible light, microwaves or X rays are a way of "projecting" energy from a point to another.

An easy example is radio waves.
You have an antenna connected to a resonant circuit where basically electrons are forced to go back and forth (along the antenna): this produces a EM wave that carries the energy used to move the electrons through empty space to another antenna where electrons pick up this "radiant energy" and start to move as in sympathy with the first...BUT...to start everything you need a generator (or an energy input) to start the movement of the transmitter's electrons!

So, your wave is "simply" a way that energy has to pass from point A (the transmitter) to point B (the receiver), a kind of....a cab for energy!!!

If you are interested in energetic considerations you could have a look at Einstein's Relationship relating the Energy carried by a photon of light to frequency: #E=hf#

If you are even more REALLY interested have a look at the idea of Poynting Vector...you'll find it quite interesting!