the 4 diodes in full wave bridge will convert the positive and negative parts of AC waveform into a positive only waveform that has both of them bouncing on the positive side. to convert it to a clean dc signal, you need a large capacitor which will take care of all of the ripples and will create a clean dc signal. if you only used two diodes (half wave), it just cuts the ac signal's negative off and only gives out the positive pulses from the ac. if you look at the waveform through an oscilloscope of a half wave bridge, where the negative pulses were at, it's just ground/0v.
easier way to think of it is full wave bridge's waveform is "bump bump bump etc" while half wave is "bump nothing bump nothing etc".
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