Dave Cornutt offers a nice overview of how a VCO works:

Synth circuit designers, starting with Moog and Buchla in the early 1960s, long ago turned to “artificial” oscillator circuits that do not rely on natural resonance. Rather, they rely on an analog implementation of a mathematical function that can be made to increase and decrease proportionally to the control voltage. A second consideration is that a useful synth VCO is expected to output several different waveforms, so that the synth player has harmonic starting points for creating a given sound. Rather than create a separate VCO circuit for each desired waveform, it is far more cost-effective to design a VCO “core” that generates a waveform from which the other desired waveforms can be derived.

At his site, he goes into considerable detail and has illustrations to expand on his explanation.