Christ's atonement as a sacrifice deals with the guilt of our sins and the atonement as a propitiation means that God’s wrath is now removed from us. So the righteousness of Christ now covers over us so that God does not look at us in wrath but in pleasure. John Murray writes:
Propitiation presupposes the wrath and displeasure of God and the purpose of propitiation is the removal of this displeasure. Very simply stated the doctrine of propitiation means that Christ propitiated the wrath of God and rendered God propitious to his people. (John Murray, Redemption: Accomplished and Applied, Grand Rapids: William B. Eerdman's, 1955, 30).We can see this in Paul’s letter to the Romans.
But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it—the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: for all have sinned and fall short of the glory of God, and are justified by his grace as a gift, through the redemption that is in Christ Jesus, whom God put forward as a propitiation by his blood, to be received by faith. This was to show God's righteousness, because in his divine forbearance he had passed over former sins. It was to show his righteousness at the present time, so that he might be just and the justifier of the one who has faith in Jesus. (Rom. 3:21-26)Here Paul is trying to explain how God can justify sinners without becoming a liar through declaring what is unrighteous to be righteous. So the question has to be how sinners are actually made to be righteous in the sight of God. It cannot be by their own obedience to the law as no one perfectly obeys God’s law. Instead all are sinners when judged by God law. But the Gospel says that a righteousness that is from God apart from the law has now appeared. So we are all justified by God’s grace through Christ who is our propitiation. So God remains righteous because as we are redeemed in Christ we are found to be just in his sight, not by any lie or deceit on God’s part, but because we are covered by Christ’s righteousness. By this we see that God is able to be consistent with his own character in justifying sinners. Again, John Murray writes:
It is one thing to say that the wrathful God is made loving. That would be entirely false. It is another thing to say the wrathful God is loving. That is profoundly true. But it is also true that the wrath by which he is wrathful is propitiated through the cross. This propitiation is the fruit of the love that provided it. . . . God appeases his own holy wrath in the cross of Christ in order that the purpose of his love to lost men may be accomplished in accordance with and to the vindication of all the perfections that constitute his glory. (Murray, Redemption, 31-2).
This all helps to emphasize the gracious nature of God's work to redeem us in Christ. In order to appease his own wrath the punishment had to be paid and this was done by Christ's sacrifice. In order for us to be righteous we had to possess perfect compliance with all of God's demands in the law. That is done by Christ's propitiation so that God looks upon us as righteous and Christ's and loves us as those who are obedient to him in the Son.