the Ramban (Vayikra 1:9) presents what he considers the most compelling rationale for korbanot: since human action is completed through thought, speech, and deed, the sinner brings an animal and lays his hands on it corresponding to deed, confesses verbally corresponding to speech, and burns the innards and limbs — the organs of desire and action — corresponding to thought and the limbs of the body, so that he recognizes he deserved to have all this done to his own body and soul in place of the animal.
the Ramban (Vayikra 4:2) deepens this by explaining that every sin produces a blemish on the soul, and the soul cannot stand before its Creator except in purity, so even one who sinned inadvertently brings a korban in order that the soul may draw close again to the God who gave it — which is why the pasuk pointedly uses the word "nefesh."
the Ramban (Vayikra 17:11) adds a dimension grounded in the blood itself: God gave us the blood of animals to be on the altar to atone for our souls, so that the animal's life-force serves as atonement for the human being before God — for all souls belong to God, and it is the neshamah of the animal, not its flesh, that is fitting to be brought before Him.
the Ramban (Bereishit 4:3) grounds this entire institution in primordial human knowledge, noting that Kayin, Hevel, and Noach already understood "a great secret of the korbanot," which silences those who are troubled by the concept, pointing the reader toward the fuller exposition at Vayikra 1:9.