Aslan has a long talk with Edmund, and while no one hears what he says, Aslan's words clearly have a positive effect
Aslan admits that the Witch's words are correct
there are forces that even Aslan cannot fight
although he may be the god of Narnia, even he must answer to a higher law.
Instead, like Christ, he sacrifices himself to atone for another person's sin.
Similarly, in The Lion, the Witch, and the Wardrobe, Aslan cannot fight the law that traitors must forfeit their lives to the witch, but he can sacrifice himself in Edmund's place. Aslan's sacrifice must be what Aslan and the Witch discuss.
Lewis builds anticipation, describing only the reaction of Aslan and the Witch to their agreement.
We also realize that he must love Edmund intensely for him to make such a sacrifice to save his life.