Its very simple. You just have an input coil too (basically a pickup) that picks up the signal from the string. The signal is amplified by an opamp and the output feeds a driver coil (which excites the string). If you invert the signal to the driver coil, it damps the fundamental, which pulls the harmonics out of the string (aka "harmonic mode"). The real problem is avoiding unwanted feedback.
Here's a schem of a simple one:
The Patent:
http://www.diyguitarist.com/Misc/Patent_4075921.pdf
[EDIT: Gearhead beat me to it]