Why do sacrifices please and appease God?

Your instinct is partly right, but I want to correct where the weight of your thinking falls. You locate the value of a sacrifice mainly in what the offerer gives up, or in the virtue he shows. That is not the main reason sacrifice appeases God. Let me lay out the traditional Catholic understanding, following Aquinas, Augustine, and the Fathers. First, understand what a sacrifice properly is. As Aquinas says, quoting Augustine, "a sacrifice properly so called is something done for that honor which is properly due to God, in order to appease Him." A true sacrifice is a good work done so that we may cling to God in holy fellowship. Sacrifice is at bottom an act of homage, an offering of honor owed to God. That is why it appeases. Just as a man overlooks an offense committed against him "on account of some pleasing act of homage shown him," so God is appeased by an act that pleases Him and is referred to Him in charity. Second, and here your instinct needs correcting, the appeasing power of a sacrifice does not rest chiefly on how much is lost or endured. Notice what Maldonatus said against Calvin: it was NOT the greatness of Christ's punishment that satisfied God, "but the condition of his person which satisfied God." Christ's Passion was a true sacrifice precisely because "this voluntary enduring of the Passion was most acceptable to God, as coming from charity." The value lies in the charity, the obedience, and the dignity of the one offering, not merely in the suffering or the renunciation. Suffering by itself, without charity and without being referred to God, appeases nothing. Third, there is a real two-fold process here, and this answers your question of whether sacrifice accomplishes anything real. In Aquinas's view: (1) to begin redemption, God must be appeased (propitiation), and (2) once appeased, He infuses grace, which cleanses the individual from sin (expiation). This is why Hebrews 9:22 says, "Without the shedding of blood there is no remission of sin." A proper sacrifice is not accepted for no reason. It honors God, renders what is owed, and opens the channel by which grace flows to cleanse the sinner. Now, to your ordinary sacrifices: fasting, almsgiving, penance. These please God for the same underlying reason. They are good works done in charity, offered to God as honor, uniting us to Christ's one sacrifice. The renunciation of worldly things and the virtue involved are real and they matter, but they are the material of the offering, not the source of its power. The source is that the act pleases God, is made in charity, and is referred to Him. A "sacrifice" with the outward loss but empty of charity and honor toward God, as with Cain, does not appease. So you are right that a sacrifice is valuable for what it implies, but be precise: what it must imply above all is charity, obedience, and honor rendered to God, and in Christ's case the infinite dignity of His divine Person. One caution about the framework itself: atonement is not a merely human scheme to placate an angry deity. As I have written, atonement is God's own design, since He requires supreme sacrifice to preserve His integrity as He deals with sinful men. God is bound not by contract but by His own righteous and holy character. Works cited: - CASB 1 - Matthew - Second Edition 2019 - Commentary Vol 1 05.02.22 - Bible Studies for Catholics - Hebrews 9:22 - Romans 11:22 - Wisdom 14:9 - Psalm 5:7