Most VCO circuits do no produce all of their output waveforms independently. Rather, a part of the circuit called the core produces one waveform, and the other waveforms that the VCO produces are derived from the core waveform. One such core is the sawtooth core. It works by converting the control voltage to a current, and then using that current to charge a capacitor, usually referred to as the integrating capacitor. As the capacitor charges, the voltage across it increases at a rate proportional to the control voltage, which is what produces the rising (or falling) ramp of the sawtooth wave form. When the voltage across the integrating capacitor exceeds a reference voltage, a comparator detects this and triggers a circuit that instantly discharges the capacitor, producing the "dropoff" portion of the sawtooth, and then the cycle repeats. With this setup, the charging current determines the rate at which the capacitor charges, which in turn determines the output frequency.
The sawtooth core is the most common VCO core design. One well known alternative design is the triangle core.