It is easier to understand the moral influence account if it is read first as an objection and only then as a proposal, because that is the order in which Abelard states it.
The objection is a question about the character of God. If God is willing to forgive, Abelard asks, what stands in the way? To say that God cannot forgive until an innocent man has been killed is to say that the death of the innocent is what makes God gracious, and Abelard held that this makes matters worse rather than better. He puts the point sharply: it seems cruel and unjust that anyone should demand the blood of an innocent person as a price, or be pleased by the killing of one who was guiltless.
That objection is not answered by pointing out that Christ consented, because the question is about what the arrangement says of the one who required it. Nor is it disposed of by observing that Abelard had another theory to sell, since the objection would stand even if he had none.
The proposal follows from it. If the obstacle is not in God, it is in us. The cross is then God's act of displaying a love that goes to the furthest point, and its effect is that those who see it are drawn out of fear into love and freed to live differently. Abelard describes the result as a person bound to God by love so that they do all things out of love rather than out of dread of punishment.
Two features of this deserve emphasis because they are easily lost. Abelard does not say that God is unaffected by sin, and he does not drop the language of redemption. What he denies is that anything had to be paid to anyone before God could forgive, which is a narrower claim than the one his critics usually attribute to him.