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Anselm's understanding of the Atonement iirc is that we owe God, as he is our Creator and King of kings, honor, which is shown through obedience. Since we have sinned, there is a debt of honor that has to be paid. We cannot pay this back ourselves since any good deeds we do are already owed to God. It was the primary understanding of the Atonement in the RCC until Aquinas, who popularized the idea of the Atonement as Recapitulation and penance for original sin.
Interestingly, the difference in Anselm's and Aquinas's takes on the Atonement play a major part in the differences between the RCC and Lutheran groups. The former held to Aquinas's view, with its implications of a more developed penance system for "paying off" mortal sins with penance, while Lutherans, following Anselm's understanding, saw any good deeds as being owed to God anyway, and so dispensed with the developed penance system while retaining private Confession and Absolution with a focus on repentance.
Both are of course very different than the "Christus Victor" approach often seen in the Eastern Church.