But now apart from the law the righteousness of God has been made known, to which the Law and the Prophets testify. This righteousness is given through faith in Jesus Christ to all who believe. There is no difference between Jew and Gentile, for all have sinned and fall short of the glory of God, and all are justified freely by his grace through the redemption that came by Christ Jesus. God presented Christ as a sacrifice of atonement, through the shedding of his blood—to be received by faith. He did this to demonstrate his righteousness, because in his forbearance he had left the sins committed beforehand unpunished— he did it to demonstrate his righteousness at the present time, so as to be just and the one who justifies those who have faith in Jesus. - Romans 3:21-26
The phrase in bold actually says that "god set forth (pro-te'-tha-me) Christ to be a propitiation (he-lä-sta'-re-on) through faith in his blood."
Faith in his bloody sacrifice as a fulfillment of the Atonement day ritual is what leads to righteousness with god. It was on that day that the High Priest confessed the sins of the nation over the head of a goat and then cut its throat and offered its blood in the Most Holy. A second goad, the goat for Azazel or scapegoat, was chased off into the wilderness.
It was not simply that god knew the method that the Romans would use to kill Jesus, the shedding of Jesus' blood was an essential requirement for the forgiveness of sin.
Now lets take a closer look at the two Greek words I have highlighted above.
According to Strong's lexicon pro-te'-tha-me means "to set forth to be looked at, to expose to view; to expose to public view; of the bodies of the dead"
It was common practice in the ancient world to leave the bodies of executed criminals exposed to public view as a warning to others. Paul uses precisely this imagery in describing what god did to Jesus. It was not the Romans or the Jews who made a public spectacle of Jesus' broken and bloody body it was god himself.
Further, according to Paul, the purpose of this gruesome scene is to act as a propitiation - he-lä-sta'-re-on. Again according to Strong this word was used of the cover of the ark of the covenant in the Holy of Holies, which was sprinkled with the blood of the expiatory victim on the annual day of atonement (this rite signifying that the life of the people, the loss of which they had merited by their sins, was offered to God in the blood as the life of the victim, and that God by this ceremony was appeased and their sins expiated); hence the lid of expiation, the propitiatory.
In summary the picture that Paul presents is that of the wrath of an angry god appeased by the sight of Jesus bloody body hung on the cross.