Hi Ray,
Simple explanation, excuse the appalling diagram.. not to scale 
In AC electricity the voltage swings either side of zero volts to give an average swing each side of zero of the quoted value. (A)
The rectifier turns this into a series of swings all one side of zero, and now measuring it will give you the same average voltage. (B)
Add a capacitor and it 'pumps up' to the maximum extent of the swing (which is about 1.414 times the average value) and as it has nowhere for the charge to go, the voltage stays at that higher value. (C)
Put a load on the output and the voltage will drop to the point where the charge 'pumped in' at the peaks will balance that taken out by the load over a complete cycle.
Transformers also usually give more than their rated voltage when the load is light, the %v drop between offload and on load voltage is called the transformers 'regulation', typically 10% for a reasonable one, as high as 25% or more for a very small one.
The role of the capacitor is to smooth out this lumpiness, and this is why it's best for them to be rated at at least twice the AC voltage.
Neil
