Let's use Leviticus 1 as an example:
The burnt offering was an unblemished male. The offerer (lit. one bringing the offering) placed his hand on the bull's head so that it would be accepted on his behalf to make atonement for him. The bull was killed before the Lord. The priests offered the blood to God. The rest of the animal was prepared and burned completely on the altar.
This all points to Christ, of course, and reveals the prophetic character of the Mosaic Covenant. The unblemished male animal points to the sinlessness of Christ. The placing of the hand is to identify the animal as representing the one who offers, which we may compare with Christ's baptism in that Christ designates Himself one with sinners.
The bull is killed, pointing to Christ's death. Scripture teaches that Christ's death canceled the curse issued by the Law [Gal.3:13] by doing away with the body of sin [Rom. 6], even bringing an end to the Law by His death [Col. 2:14] (since the Law is there to reveal sin, is only for sinners, and can only apply until one dies). The animal's death signified the need for the offerer to put an end to his big problem: the guilt and corruption of sin that holds him in bondage and prevents him from living as Adam did before the Fall - in communion with God.
That the blood is offered by the priests points to Christ, who entered the Most Holy Place in the Tabernacle not made with hands to cleanse our conscience from sins to serve the Living God [Heb. 9:11ff]. Notice especially verse 22, "And according to the law almost all things are purified with blood, and without shedding of blood there is no remission." The OT blood purified and [v.13] sanctified; how much more does Christ's Blood purify and sanctify from sin, which indicates expiation [def: to make (the believer) pious by removing sin] and not propitiation [def: to make (God) favorable]. Of course, one whose sins are expiated then finds God propitious. Returning to the theme of blood, there is no remission of sins without blood. Leviticus 17:11, "For the life of the flesh is in the blood, and I have given it to you upon the altar to make atonement for your souls; for it is the blood that makes atonement for the soul." Here the blood, offered to God (but only by God's commandment), is really God providing blood for us that we may return to Him and remain in communion with Him. Hence God says, "I have given it..."
Taking into account all that has been said thus far, the blood makes atonement in that it sanctifies and purifies; it provides what is lacking on man's part - which is fulfilled in Christ by His offering of perfect obedience and perfect communion with God. This Divine-Human Christ alone is able to supply to us what we lack in order to return to God: both an end to our bondage to sin, death, and the devil, AND re-entry into the Kingdom of God in perfect communion, obedience, righteousness, holiness, etc.
That the rest of the animal was offered to God by consumption through fire indicates the total and complete nature of the offering. As the animal represented the offerer, it signifies that the offerer was not to be double-minded about his turning towards and ascent to God, but was to be completely sincere, holding nothing back.
In this there is no sense that God was punishing anyone or demanding to be satisfied by someone's punishment. He was demonstrating man's need to be freed from the demonic kingdom of sin and death, and He was demonstrating man's need for an unblemished life (morally and ontologically) in communion with Him.