Atonal sound aside, this electric guitar and Tesla coil are a match made in heaven. The guitar can be hooked up to the coil because both rely on the principal of "electromagnetic induction." It's like this: when a changing magnetic field passes inside a closed loop of conducting material like wire, an induced electric current will flow in the wire. Instead of the energy coming from a battery it comes from the magnetic field. (This is how a generator operates. You spin a coil of wire inside some magnets, and the constantly changing field within the coil results in an electric current.) The guitar and coil each employ electromagnetism; pair them up and the sparks fly.
In an electric guitar those special guitar strings actually have little magnets in them. And the "pickups" located directly underneath the strings on the body of the guitar have tiny coils of wire inside. When the magnetic string vibrates it induces a current in the pickups that depends on the frequency of the vibration. This electric current then flows through a circuit which contains an amplifier/speaker component. The speaker converts the electrical signal into that rock and roll (sound) that we know and love.
In this case, however, that circuit is extended to include a lightning bolt-spewing Tesla coil. Some of the specifics of the way Tesla coils work can get complicated but basically they contain coils of wire (called inductors) and devices that store charge (capacitors). Using EM induction, these inductors and capacitors affect each other in such a way that an electric current in one coil is able to induce a very high-voltage current in an adjoining coil. It's the high voltage that gives rise to those freakish lightning-like discharges.
Even better than the bolts is the way they follow the pattern of the music. Since the irregular electrical signal produced by the guitar is driving the Tesla coil, it stays in rhythm.
Big Tesla coils achieve millions of volts, and result in discharges that can kill you. With the mere hundreds of thousands of volts in action here, a flyby spark would probably cause just a painful jolt accompanied by an unpleasant tingling sensation—and maybe a headache. Rock on!
Adam Weiner is the author of Don't Try This at Home! The Physics of Hollywood Movies.