What does this circuit do? Answered

A fluorescent tube produces light by forming a plasma which emits ultraviolet light, this produces visible light by inducing fluorescence in the white-coating inside the tube,
To produce the plasma high voltages are required, and this is what the circuit does. The little black transistor and the transformer convert low voltage DC to HV.
You could just connect the thing to a HV cap', provided that the voltage is appropriate.

L

I believe this circuit produces a AC output so you would need to rectify it to charge the capacitor. I expect that a 1N4007 diode would suffice but it would be safest to check the voltage output from the transformer first.