Francis Turretin (d. 1687) did a very good job of summarizing Scripture’s account of Christ’s atonement as satisfying the justice of God. In his section called “The Necessity of the Satisfaction,” he wrote the following (I’ve edited it slightly):
Sin, which renders us guilty, binds us over to punishment as hated of God. It [sin] may be viewed 1) as a debt which we are bound to pay to divine justice, in which sense the law is called “a hand-writing,” (Col 2:14) 2) as a principle of enmity, whereby we hate God and he becomes our enemy: 3) as a crime against the government of the universe by which, before God, the supreme governor and judge, we become deserving of everlasting death and malediction.
Whence, sinners are expressly called 1) “debtors,” (Matt. 6:12); 2) “enemies to God,” both actively and passively, (Col. 1:21); 3) “and guilty before God,” (Rom. 3:19.) We, therefore, infer that three things were necessary in order to our redemption; the payment of the debt contracted by sin, the appeasing of the divine wrath, and the expiation of guilt.
[Therefore] the nature of the satisfaction to be made may be easily perceived: the payment of the debt, the appeasing of the divine wrath (by reconciling us with him) and the expiation of guilt (by the endurance of punishment).