why there is APU bleed air supply?
The APU is an alternate source of electrical power and also bleed air.  Since it doesn't have any function to generate thrust, it burns considerably less fuel than the engines do, and is intended primarily for ground use, although most APU's can also be operated in flight for things like a second source of electrical power if an engine generator has failed, or a source of bleed air if the bleeds are closed on the engines in order to generate more thrust.

In that diagram, what you're seeing is a portion of the pneumatic system, showing that the bleed air from the APU can be routed to both air conditioning packs (on many aircraft the APU can power both AC packs on the ground; on some aircraft it can only power one at a time), as well as to the engines, where it provides air for starting them.

On some aircraft, the bleed air from the APU can do other things, like pressurizing the water supply, and while it isn't operationally useful, it can generally supply bleed air into the anti-ice ducting for the leading edges.  In practice, though, that's only done when the engines are running, due to the greater volume (and temperature) of hot bleed air that the engines can produce compared to the APU.