This lamp takes (2) bulbs correct? That would explain the two chokes.
I can't say for sure from the picture but those caps really do look like they are just for noise suppression. I have a flexarm desk lamp without these caps and it will make a "blip" on the TV or radio when the lamp is turned on. It is also possible that the caps are for power factor correction but I have my doubts. I would be surprised if they affect the way the bulb works at all.
The main purpose of the choke is to regulate current. A fluorescent lamp does not have any resistance and much like an LED needs a constant current supply. You can think of it sort of like a big resistor. The choke cyberdyke suggested is a good option if yours are bad. The tester you are using might require more current that what the chokes will pass. Even if yours are good the tester may not light or light dimly.
One question, when you had the lamp put together and tried to test it. Did you hold the red button for several seconds? That is necessary to preheat the bulb.
As for the operation, somewhat simplified. This is how pretty much all of these desk lamps work. Yours will have double of some parts since it has two lamps (bulbs).
First you press the red button. Current flows though the filaments warming up the lamp.
After a few seconds you release the button and the sudden stop of current though the choke makes an "inductive kick" (google if you don't already know). That combined with the warm lamp is enough to start an arc across the lamp. Current flows though the lamp and there is light!