Hi,
First suppose that both capacitors are discharged.
In the positive cycle we have :
D1(first diode from left)=off, D2(second diode from left)=on
therefore both capacitors are in back-to-back position and C2(the output cap) prevent a large DC current to flow through
C1. As a consequence, C2 prevents C1 from damage!
Anyway, in the positive cycle, C1 is charged to a voltage equal to the peak of the input voltage (Vm) (transformer secondary voltage peak).
in the negative cycle we have:
D1=on , D2=off
in this cycle C1 is charged to a voltage equal to Vm.
now, in the next positive cycle ( when D1=off and D2=on) C2 receive a charging voltage equal to 2Vm from secondary of transformer and C1.
now you have an output DC voltage which is equal to twice of the transformer output voltage peak.