If the RMS value of a signal is zero, the signal does no work.
The reason you use RMS voltage rather than mean voltage for a signal is that both negative and positive voltages can do work. So instead of just taking a plain average, where negative and positive components of the signal can cancel out (and you underestimate the amount of work done), you first square the values of the signal.
(Keep in mind, however, that the RMS value of any signal can be calculated: not just voltage signals. It is a good way to quantify the deviation of a sampled or analytical signal from zero.)
This way, when you take the average, any deviation from 0 volts contributes to the total RMS value. Taking the square root of this voltage value allows you to represent the final quantity with units of volts so you can relate it to current and impedance just like in DC circuit theory.
For a sine wave (which goes from -1 to 1 and has an average value of 0) you would first square the signal. This results in a wiggly looking thing ranging from 0 to 1 who's average is one half -- the sin^2 curve. Taking the root of this average gives you 2^(-1/2) or .707 -- the RMS value of a sine wave.
This value is useful mostly in electrical engineering and physics.