This has nothing to do with the bridge rectifier, the same would apply to just the voltage source. It is about the relation between the amplitude of the sinusoidal signal (peak height) and the RMS voltage. Note that amplitude indicates the V
peak, which is half of V
peak-to-peak. V
rms is the root of the mean of the squared voltage, or the square root of 1 over the period times the definite integral of the sine squared over the period (see the bottom equation for V
rms in
this Wikipedia entry). If you go through the calculus, you'll find that the result is that V
rms = 1/sqrt(2) times the amplitude of the sine. This is where the sqrt(2) factor comes from.
Now draw the sine input (indicate zeros and maxima/minima) and the output from a half bridge rectifier (leave away the smoothing cap). Indicate over which portions of the wave the diode is conducting (forward biased). Do the same for a full bridge rectifier. How does the peak amplitude of the output of the rectifier relate to the RMS voltage? What is the difference between the output from a full and half wave rectifier?