What's the purpose of sacrificing animals in Jewish tradition/law?
Read Quoth Rabbi Kornfield's article on THE SACRIFICIAL SERVICE:
Although this practice may seem bizarre to the uninitiated, the early commentaries point out the profound and enduring effect that offering a sacrifice has on a sinner. Man, like animal, is a physical organism of flesh and blood. Both are driven by their lusts and emotions; all that separates man from animal is his intellect. When a person sins because he allows his desires to get the better of his intellect, he puts himself on equal footing with the animal. It is necessary to impress upon such a person the futility of physical existence which is not led by the mind's rational judgment. This is accomplished by having him offer a sacrifice.
The sinner brings a body of flesh and blood, like his own, to the Mikdash. There, he slaughters it and burns it on the altar -- much as he allowed his own body to be "burned" by the fire of desire (Rabeinu Bachye 1:9). The blood is sprinkled on the altar and the intestines and fats (the vehicles of desire) are burned, to demonstrate "there, but for the grace of G-d, go I" (Ramban 1:8). It is thus vividly portrayed to him that a body without a mind is valueless (Chinuch, Mitzvah #95). The meat of the sacrifice provides sustenance for the Kohanim, educators of the people and purveyors of the Torah (Devarim 33:10), who then pray for the complete atonement of the sinner (Ramban).
There is also an idea, found in the Zohar and later kabbalistic and chassidic literature, that each of the major types of animal life "evolves" spiritually from the corresponding "face" of the Chayos Hakodesh (a type of angel described in Ezekiel ch. 1 passim). The Chayos are described as having four "faces": a lion, a bull, an eagle, and a human being. The lion's face, then, is the spiritual source of all wild animals, the bull's face of all domesticated ones, and the eagle's face of all birds.
When we offer a sacrifice, the physical substance of the animal or bird goes up in smoke. This represents how it is spiritually elevated and goes to "feed" the corresponding "face" of the Chayos, thus bringing down more Divine largess to that category of creature (and thereby to us, who depend on them for food and other uses).
Rambam - In order to fight our urge for Avoda Zarah
Ramban - To bring one closer to Hashem