Stated by those who hold it, the argument runs in four steps.
First, the problem is described. Sin is not merely damage to be repaired but wrongdoing to which a penalty attaches, and the penalty is not an arbitrary attachment but the just consequence of the wrong. A God who simply waived it would be indifferent to what had been done and to those it was done to. On this reading, the seriousness of the penalty is a measure of how seriously the wrong is taken, and any account without it makes light of both.
Second, the texts are read. Defenders point to the servant of the fourth song in Isaiah, wounded for the transgressions of others and bearing their iniquities; to the statement in Galatians that Christ became a curse in our place; to the exchange in the second letter to the Corinthians in which the sinless one is made sin so that sinners might become righteousness. They argue that substitution is the plain sense of these, not an inference imposed on them.
Third, the mechanism is stated with care, because the most common objection turns on a misstatement of it. Defenders insist that this is not the Father punishing an unwilling third party. It is God, in the person of the Son, taking on himself what was owed by others, and the willingness is the Son's own. On this account the Trinity is not divided at the cross; God bears in himself the cost of forgiving.
Fourth, the result is drawn out pastorally, and this is where the doctrine has most of its practical force. If the penalty has been borne, then it will not be exacted again, and the assurance available to a believer does not depend on the state of their feelings or the quality of their repentance. Preachers in this tradition regard that as the doctrine's chief gift and the main reason they are unwilling to trade it for an account in which nothing definite was settled.
It is confessed in the Reformed and Lutheran standards and in the confessions descended from them, and this is the shape those documents give it.