I'm not sure, and without a circuit diagram it's difficult to be sure, but there are a few clues from the pictures, and it seems to follow the basic idea of page 9 of
http://www.alanwood.net/downloads/olympus-bh-2-bhs-repair-manual-lq.pdf, but really just the top part of that diagram. In that, UYPC64 is the power controller, and I can't see any equivalent of that in your circuit. Your circuit doesn't seem to have any equivalent to the transformer with the 7 V winding or the UYPC65 section.
It seems that the power goes through the switch and then the fuse. I think that the power then goes through the filter (the "Don't know what this is")
The power is then somehow controlled. I don't know how. The two variable resistors, the round one and the slider, are probably not capable of taking all the current, so there is probably a triac or something we can't see. The missing bit would be vaguely equivalent to the UYPC64 part of the diagram.
The power then goes to the inductor, the rectangular component with the red and the black wires. Then the power goes to the toroidal transformer, the large round orange thing. The other power lead from the switch is white, and it goes to the voltage selector, (round black thing with 4 green blobs) and then to the toroidal transformer. The output of the toroidal transformer is on the right, near the mains input, and goes to the lamp and also to a large red resistor. I guess that the resistor is there to give some load if the lamp blows.
I don't know if this helps, and it doesn't really answer the question as to what the capacitor does, apart from probably being part of the circuit that controls the voltage to the transformer.