What is the use of transformers with 3 pairs of wires?
You are looking at the transformer of a very elegant and basic resonant converter (a.k.a. Royer converter).

A very basic explanation is this: There are two windings with an equal number of turns that are used to drive two transistors, and there is one winding that connects to the lamp. During each switching cycle, at a certain current, determined by the core material and the number of turns of the winding that goes to the lamp, the transformer will saturate and thereby define the power that is transferred to the lamp. Each time a saturation event occurs, one transistor will block and the other transistor will start to conduct, which keeps the circuit oscillating.

The Royer converter was invented long before switch-mode power supplies were the norm for power conversion circuits and is still very popular. Considering its age, it is amazing that efficiencies of > 85 % can be achieved easily.

A good application note with many details can be found here: http://www.nxp.com/documents/application_note/AN00048.pdf.

There is also an article on Wikipedia: http://en.wikipedia.org/wiki/CCFL_inverter

Another good App'note: http://www.diodes.com/zetex/_pdfs/3.0/appnotes/apps/an14.pdf

Here's a good manual about fixing compact flourescent lamps (CFLs). It includes many schematics: http://www.en-genius.net/includes/files/col_081307.pdf

The original reference to the circuit is here:
Bright, Pittman and Royer, “Transistors As On-Off Switches in Saturable Core Circuits,” Electrical Manufacturing, December 1954.