We used to work on projects with electrostatics all the time when I was in engineering grad school. Basically, it's like playing with two magnets with a "diaphragm" between the magnets. Instead of magnets, the speakers or headphones in this case use "stators", which is a fancy term for the transparent grilles. The grilles are electrically conductive material, but typically acoustically "inert" as well. Thus, the stators are conductors but don't color the sound acoustically. Between the stators is inserted a material used as the diaphragm. In school we used all sorts of material but I suspect the typical source is Mylar, very rigid, durable etc.
The stators are enabled with current conducting materials throughut, usually copper, but not always especially in headphones as I understand it. The amplifier simply delivers voltage to the stators and the diaphragm. This creates an electromagnetic environment where there are positive and negative charges just like a magnet. The positive charges repel one another just like sticking common ends of a magnet together while the negative and positive attract one another.
A signal passing through the amplifier creates a positive voltage on one stator and a negative voltage on the other stator. Then, the voltages alternate back and forth between positive and negative but at ultra high speed of course.
This sound is prouced by the following. At one instant, the front stator will be positive while the back is negative. The diaphragm's polarity does not change and remains static. If the diaphragm is negative it will be attracted to the positive stator and repelled simultaneously from the negative stator. Remember, the diaphragm does not change polarity like the stators.
The amplifier poduces current in costantly reversing polarity. The constantly reversal of the stators polarity changes the polarity of the stators which in turn causes the diaphragm to move the opposite of the way it moved in the prior signal impulse. This "oscillation" of course happens very, very quickly and produces air pressure waives through the movement of the diaphragm, just like any speaker or headphone. The ear perceives the waves as sound or music.