There are several types of capacitor-type stud finder, see the excellent Wikipedia article for more information.
The basic principle is that the capacitance of a capacitor of fixed geometry is proportional to the dielectric constant of what is between the plates. The drywall is a constant and can be calibrated out, leaving the difference between wood (dielectric constant of maybe 2 or 3) and air (dielectric constant of 1).
The seminal patent was US 4099118, which discloses a 3-plate capacitor design allowing the edge of the stud to be detected by differential capacitance.
That's the sensing principle. There are a number of ways of measuring capacitance, often making a ramp or an oscillator incorporating the unknown capacitance and measuring the time for a given voltage change or the oscillator frequency is an easy way, but there are others.