Well such a system has two major components. First is a dual resonant transformer. Dual resonant transformers have a secondary with hundreds of tightly packed windings which is self resonant at some frequency (usually at least a few megahertz, for a small coil). The primary will be somewhere between two and ten turns, and is loosely coupled to the secondary. A capacitor is normally put in series with the primary in order to tune out some leakage inductance. The function of the transformer is to step up the voltage of the driver to very high voltage. The tuning of dual resonance transformers is in itself a strange art. There is plenty of math out there used to describe it, but in general it turns out being a trial and error process (but it's absolutely necessary for good results).
The second major part of the system is the driver circuitry. There are a few different topologies that can be used, such as tube amps, class E amps, royer oscillators, etc. If I were to build one I would probably go for a class E amp (less issues with stability). Also a class E amp can be driven with a digital source. However, getting a powerful amplifier is itself a difficult process.
To make the arc play audio, it's just a matter of modulating the power delivered from the amplifier. This is commonly done by controlling the supply voltage to the amp (but can also be done with PWM).