The peace offering names how a person restores communion after conflict or good fortune. A man cannot simply feel grateful and expect the debt of relationship to settle itself. He brings the very thing that matters to him and lays it on the altar, in the open, where others see what it cost him. Reconciliation asks for substance, not sentiment.
The portion here was shared, eaten together, so the bond became visible and mutual. Peace between people gets built the same way. You mend a torn relationship by giving something from your own store, and by doing it where the other person can witness the sacrifice.