The accessory gearbox is wrapped around the front section of a modern turbofan. It includes things like the electrical generators (to supply the aircraft electrical systems) hydraulics pumps to power flight controls, the engine oil sump and pumps and start motor. On the ground, the engine is started using typically an electrical motor for smaller engines or an air start nmotor, which is like a car turbo, which takes high pressure apu air and converts it into rotation. Tis rotation is fed, via a clutch, into the engine gear box via a shaft.
THis then starts the engine turning. Once the engine is rotating fast enough, usually at around 20% core speed, fuel is introduced. The air that is being sucked through the engine is used, the engine does not use the apu air for combustion. The starter typically disengages once the engine is 'self sustaining', his usually occurs at around 50-60% core speed.
Wikipedia helps:
http://en.m.wikipedia.org/wiki/Accessory_drive
That image you have on is a much simplified version of a turbojet,it is good for explaining how a jet works but jot much else!