The gods traditionally received the thigh bones of the sacrificed animal wrapped in fat, as well as some of the internal organs. If a priest was involved, he too had claim to some of the organs. Who exactly got what innards varied by time, place, and deity. For example, priests usually got the tongue of the animal, unless they were priests of Hermes, since Hermes demanded the tongue as part of his due. It was also not at all uncommon for the sacrificer to forfeit the hide of the sacrificed animal to a temple (the temple being supported in part from the proceeds derived from selling the hides).

So thigh bones, fat, a collection of organs, maybe a hide: no matter how you cut it, this is a lean portion for a god. Even the Greeks seem to have thought so. Myths were told to justify it. When the world was young and sacrifice was first invented, it was said that Prometheus put all the flesh from a cow into one pile, and all the organs into another, carefully draping fat over them to make them look as appetizing as possible. He then asked Zeus which share the gods would take as theirs and Zeus, quite naturally, chose the good-looking pile of organs and fat. Later intellectuals who took umbrage with the notion that Zeus could be duped like a toddler said that the gods merely enjoyed the smoke and the smell of sacrifice, much as they enjoyed the smoke and smell of burning incense. Or perhaps that it was one of the many favors the gods bestowed upon men that they took the smaller share in sacrifice. The sacrificial custom probably had prehistorical roots in propitiatory sacrifices after hunts. Since the entire point of the hunt was to get food, the best foodstuff was naturally reserved for consumption.

For this and all other questions of like nature, the best starting place is probably Walter Burkert's Greek Religion. He goes into greater detail about the prehistorical roots of Greek religion as well as the division of sacrifices in the historical period.