NOTES for Lev 3:1-17
The Yahwist sacrifice, judging by the description in the book of Leviticus, included three elements. First of all, and every sacrifice began with this, came prayer. The person who wanted to offer a sacrifice laid his hands on the head of the sacrificial animal and said a prayer. In the case of the so-called sacrifice of reconciliation, or sacrifice of thanksgiving (the corresponding Hebrew word can mean both "peace" or "reconciliation" and "thanksgiving" in the sense of a return gift or payment), this was a prayer of praise or thanksgiving. Then the animal was slaughtered, after which the priest sprinkled the altar and all those taking part in the ritual with blood. The third element was the sacrificial meal itself, in which all those present also usually took part, at least when the sacrifice of reconciliation was involved (in the Synodal translation it is usually called a "peace offering"). During the meal, a separate, special portion of the sacrificial meat was intended for the priests and Levites taking part in the ritual.
This order becomes understandable if we remember that sacrifice was originally conceived as a form of communion with God and of human sanctification. By God's design, a human being is both a spiritual being and a natural one. As a spiritual being, he is the image of God, a self-aware person possessing freedom of will. Accordingly, for communion with God and for sanctification, a person must freely turn to God himself. That is why everything begins with prayer.
Without turning to God, at the level of ritual alone, sanctification is impossible for a person. On the other hand, a human being is not only spirit but also nature, animal nature, and after the fall he became much more animal than he had been before the fall. This human nature is sanctified through the sprinkling with blood and the eating of sanctified meat. By nature, after all, a person is closest precisely to animals, to those very animals God allowed him to eat after the fall.
God's presence sanctifies the animal placed on the altar, and the person, by then eating the sanctified meat, interacts with the nature sanctified by God on the bodily, and therefore natural, level. Animal nature acts on the bodily nature of fallen man, sanctifying it just as every sanctified nature acts on any other nature, sanctifying it in the process of interaction. If the fall had not happened, a person would not have needed such mediated action upon his bodily existence for its sanctification, but in the fallen state it is, as can be seen, the most adequate form of sanctifying human nature.