Further reading: http://en.wikipedia.org/wiki/Auxiliary_power_unit
Edit: after further research, it seems compressed air is not used on the 787. It's all electric (two generators on the APU and on each engine):
"The power source for APU starting may be the airplane battery, a ground power source, or an engine-driven generator. The power source for engine starting may be the APU generators, engine-driven generators on the opposite side engine, or two forward 115 VAC ground power sources. The aft external power receptacles may be used for a faster start, if desired."[1]
1. http://www.boeing.com/commercial/aeromagazine/articles/qtr_4...
Evidently adding the APU saves enough money to justify its significant weight.
- backup power for emergencies - allowing attempted engine restarts in the air - allowing the airplane to be self-sufficient should it be forced to land somewhere without ground equipment (or big enough ground equipment)