Even for purely resistive loads, power is proportional to the square of the voltage:

  P = V * I

But,

  I = V / R

So,

  P = V * (V / R)
    = V^2 / R

For chips, power scales with voltage squared. Is also true that P=IV (since both are true, these observations cannot be in contradiction). Apparently, for chips, the current must be proportional to voltage also. Glossing over some details, turning on (off) a transistor is the same as charging (discharging) a capacitor. The energy stored on a capacitor is 1/2 C V^2. If you turn on and off the transistor periodically (say with frequency f) you use 1/2 C V^2 energy f times per second (energy per unit time is power). Normally the capacitance is ignored when discussing how power changes because for a given design the capacitance is a fixed quantity.

Running a processor at higher frequency also requires increasing voltage, which increases effective capacitance by that same formula.

That's not the primary cause of the power = frequency^2 rule, but actually adds a factor on top of that.