How does light waves transfer energy?

When you rub a balloon against your sweater or a comb through dry hair, you give them electric charge, which creates an electrostatic field around them.  Then, when you pass them over bits of tissue, they lift the bits of tissue up from the table. You see that the electrostatic field must have energy in it, because it takes energy (work) to lift things up.

If you have a magnet, it has a magnetic field around it.  When you pass it over paper clips, it lifts the paper clips up from the table. You see that the magnetic field must have energy in it, because it takes energy (work) to lift things up.

Light waves ... and all other electromagnetic waves, like radio, heat, X-rays, etc. ... are a traveling combination of an electrostatic field and a magnetic field, traveling together through space at about 300 thousand kilometers per second.  The fields carry the energy that was able to pick up the bits of tissue and the paper clips.

But light or heat waves can't pick things up, because the electrostatic and magnetic fields are combined in a very special way It's the moving static field that creates the magnetic field, and it's the moving magnetic field that creates the static field.  But that's another story, that you'll learn about either in late high school or early college.