Let's start with the basics. Johnson says, in response to the question of how Anselm came up with his position:
Very simply, the way all of us come up with our ideas: from his own experience in his own world. Anselm lived in a feudal society, where there was no police force nor armies. The word of a lord was law, and this kept the civil order. If you broke a law that disturbed the order of the society in which you lived, you had to pay back something to the lord in order to restore that order. That payback was called satisfaction. You had to make satisfaction when you broke a law in order to restore the honor of the lord, on which all civic peacefulness rested.
Anselm took that political arrangement and made it cosmic.
This is a bad start, because it is all completely wrong. There's been some work in past decades showing that it is, in fact, wrong. Nicholas Cohen's Feudal Imagery or Christian Tradition? (PDF) and David Whidden's The Alleged Feudalism of Anselm's Cur Deus Homo and the Benedictine Concepts of Obedience, Honor, and Order are good examples, and develop parts of the argument at length. But we can summarize the points for our purposes here, and say that
(1) 'Feudalism' is too vague a term to do the explanatory work it is supposed to be doing here;
(2) but if we take the 'political arrangement' Johnson and others assume when they appeal to it, such political arrangements did not have practices of the sort that is suggested, which seem to be an imaginative fiction;
(3) and even so, Anselm didn't live in an area of the world that had that sort of political structure yet;
(4) and Anselm also rarely if ever uses any terms that can be given any feudal gloss at all;
(5) and when he talks in ways definitely shaped by his society, it is the society in which he actually spent most of his life, namely, the Benedictine monastery, not any secular political arrangement.
Anselm held that to be a monk was to go part of the way to being the way a human being should be; monks are not the only ones who are saved, but to be a monk is devote oneself to a life of loving God and neighbor, without being weighed down by the ways in which the world interferes with these. To love God is in part to serve and honor, i.e., respect or revere, God in His goodness, a service God does not need but that we do. And the whole of monastic life consists precisely in practices that are for honoring God. This is the model that Anselm is generally held to have in mind: we human beings can only be just by loving God, which requires serving and honoring Him. (Likewise, we can only be just by loving our neighbor, which requires honoring our neighbor, which is also a way of honoring God.) But there is a problem: we have, in fact, failed to serve and honor Him. Sin is sin because it is a dishonoring of God; this can only be fixed by restoring honor.
This is why God's forgiveness does not answer the question of how the problem of sin can be resolved -- the problem of sin is not that God has failed to forgive us but that human beings have sinned and keep sinning. Divine forgiveness on its own does not address that problem at all. Thus the problem of sin, by its very nature, requires that there be some way that leads to human beings honoring God, and to do so justly requires that we not only honor God but that, because we have sinned, we somehow honor God even more to make up for it, which is satisfaction -- the difference between unjust and just people is that the just, when they have done wrong, do what is required for satisfaction. This is where the basic dilemma in Cur Deus Homo comes from: to counter sin, for their own sake human beings have to honor God in a way that is beyond the ability of human beings to do; God alone can render honor to God in that superabundant way; so the only way we can honor God as we have to honor Him is if there is a Deus Homo, a God-Man, so that human beings are by Him so exalted that we can render the honor we owe to God and make up for what we have failed to render before. And what does Anselm say the honor we owe to God is? The service to God that constitutes justice and makes us just people. The honor we owe to God is to do what is right so that we may be truly just.
Satisfaction in a sense restores the order of the universe, but Anselm is very clear that it restores the order and beauty of the universe because it restores us so that we are just people acting in just ways. And on Anselm's account, we are not merely restored by Christ; we are exalted, so that we will become in a way the equal of the good angels. And, of course, we cannot be the equal of the good angels if we continue to sin; only by satisfaction can we be made the equal of holy beings. (In reading Cur Deus Homo, I think, people often just skip by the discussions of how human beings will complete the number of angels, but it is, I think, quite important to Anselm's actual argument, since it is his way of discussing what it means to be made holy, to become saints.)
Thus Anselm's argument is not that Jesus had to die for God to be merciful; God is always merciful. But the problem that is specifically in need of addressing is that we are sinners and need to stop being sinners. This we cannot do by except by becoming just people; but just people do not only do what is right, they make up for wrong that they have done. So divine mercy gives us a way to make up for wrong that we have done. God becomes Man, and therefore He can be a man who will honor God as we need to be able to honor God, that is, to give God the honor we need to make to Him and to give him the satisfaction for our failures to honor Him, as we also need to do. And Christ does this by giving His sinless life to God, even dying in God's honor, which latter went beyond what He Himself owed, and so took human honoring of God to the next level. What is more, it is not God who demands the death -- it's actually essential to Anselm's argument that God does not demand Christ's death, because if God did demand it, it would be owed and not supererogatory. Anselm is clear that it is human beings who decided to kill Him. But God foreknew that this would happen, and also that, because His death was made a gift to God, it would be a gift that could make up for even the sins of murderous human beings, and, further, would serve as an example of love for others to follow. And Anselm argues that a just person's satisfaction can, through the bonds of love, be satisfaction for another; so Christ's give can be ours, if we are united with Him in love.
Johnson, I think, sees herself as following in the line of Thomas Aquinas; she usually thinks of herself as a sort of Thomist in a broad sense of the term, and mentions explicitly in the interview that, "People, including Thomas Aquinas, criticized Anselm for making it necessary that Jesus do this, for taking away God’s freedom to be merciful." But while St. Thomas does deny that divine forgiveness depends on satisfaction (God's justice and mercy, unlike our justice and mercy, do not require that there be satisfaction; see, e.g., ST 3.46.2), I don't think this can actually be read as a criticism of St. Anselm, either in reality or in intent. As to intent, I think St. Thomas is really just defending an idea he finds in St. Augustine, and not particularly thinking of St. Anselm at all, although, granted, one could perhaps dispute this. But as to reality, Anselm is also quite clear that divine justice and mercy don't depend on our satisfaction -- our justice does. As Anselm at one point puts it, God had no need to conquer the devil; we did.
Johnson later notes that Anselm doesn't mention the Resurrection, which is true (although some things he says can be said to hint at it); but it's clear from both Anselm's comments at the beginning and Boso's comments in the final chapter, that the whole point of the dialogue is to argue to the God-Man as we find him anticipated in the Old Testament and depicted in the New Testament, and not to assume anything from those more than was strictly required to discuss the problem. Johnson is right that there would be no Christianity without the Resurrection, but Anselm is not explaining Christianity; he's arguing that the Incarnation is a reasonable doctrine. And far from what she suggests, it does not follow from his account that we need only one chapter from each gospel (i.e., the Crucifixion itself); it is essential to Anselm's whole argument that Christ's life was holy so that His death could be a matchless honoring of God.