The Kuzari warns us that intellectual reasoning alone will never give us the full picture of closeness to G-d which the Korbanos (or, for that matter many other Mitzvos,) gave. The Korbanos are G-d's Will - and doing His Will gives us an element of spirituality which intellect alone could never anticipate. Korbanos are a Chok, never to be fully grasped by man, and it would be tragic for man to reduce the full force of G-d's wisdom to the limitations of our own understanding. Therefore any explanation of such an esoteric concept can never do true justice to the entire picture. However, its ever present reality demands that we extrapolate the lessons that can be learnt on our respective intellectual levels of understanding.
The Ramban famously writes, that since the acts of man are comprised of thought, speech, and action, G-d commands that when an individual sins and brings an animal sacrifice, he should rest his hands upon its head, corresponding to the element of action; he should verbally confess what he did wrong, corresponding to the element of speech; he should burn the innards and kidneys that are the seat of all human thought and passion, and the animal's limbs, corresponding to the sinner's hands and feet, which carry out all of his activity; and he should cast the sacrifice's blood upon the altar to bring to mind the phrase, "his blood will be on his soul." [This is] so that the person doing [or watching] all of these actions will come to realize that he has sinned against God with his body and his soul, and that he deserves that his own blood be spilled and his body burned, had it not been for the compassion of the Creator who has accepted a substitute instead. Therefore the sacrifice atones by its blood corresponding to the sinner's blood, its soul corresponding to the sinner's soul, its limbs corresponding to the sinner's limbs; and the portions (that are given to the Kohanim) will give life to Torah teachers, who in turn will pray on his behalf.
The Chinuch adds, that the reason sacrifices are from bread, wine, and meat, is because these are central items of importance in a person's life. Therefore a person's heart is greatly affected when using these items in the process of serving G-d.
The Chinuch further explains that animals are biologically very similar to a human being - are only difference in intellect. Seeing the animal slaughtered has a tremendous impact on ones psyche, making the effects of sin very real. Burning this animal without intellect, vividly drives home the point that a person who lives without thinking will end up being nothing - like his sacrifice. At the time, one will come to appreciate his own existence and realize that he lives for a higher purpose. This visual lesson has the greatest potential to redirect a person away from sin.
One might think that for the Peace Offering the above explanation is not applicable, and therefore why should it be a form of sacrifice? The Chinuch beautifully explains that it is a Mussar lesson about life, how to be grateful, and focused on doing the right thing. By bringing such sacrifices, a person inculcates within himself the right perspective, thus potentially preventing him from sinning in the first place. The harmonious existence indicated by the dividing the parts between G-d, the Kohanim, and himself, imbues him with the lesson of maintaining the right balance between the spiritual and the physical, and that it all serves to bring one closer to G-d.
The Prophet Hoshea says, "Take words with you, and turn to the Lord. Say to Him: forgive all iniquity and receive us graciously, so we will offer the words of our lips instead of calves." Although we may not have the enhanced ability to learn the above lessons, as it was in the time of the Beis Hamikdash, the prophet reveals to us, that by reciting the Service of the Sacrifices, one can accomplish the same even in our present state. May we merit the day soon that only offerings that will be brought will be the "Peace Offerings," indicating the peaceful and harmonious relationship between the Jewish People and G-d.