According to Jones, this theory is developed by anthropologist and literary critic, René Girard. Jones quotes from James Alison in an article published here: http://www.jamesalison.co.uk/texts/eng05.html, which describes Girard's way-of-thinking about humanity. Basically, he says that we all have a desire for what other humans have. This idea is called "mimetic desire." We want to be like others, and have what they have. Girard posits that this mimetic desire leads to rivalry and violence as we try to get what others have. This rivalry and violence grows and grows rampantly, until the community unites against a Scapegoat who takes on all of the guilt and is sacrificed. (I'll admit, that's probably not exactly the best explanation, but that's how I re-phrase it in my own words to try and explain it to you).
Much of human religion was sacrificial religion until Judaism and Christianity changed things and wrote the story differently. Ultimately, Jesus Christ is the Last Scapegoat. Here's what Jones writes:
"In Christ, God becomes the one who is rejected and expelled. That is, the scapegoat is not one of us who is sacrificed to appease an angry deity. Instead, the deity himself enters our society, becomes the scapegoat, and thereby eliminates the need for any future scapegoats or sacrifices."
He then quotes James Alison's summary: "Christianity is a priestly religion which understands that it is God's overcoming of our violence by substituting himself for the victim of our typical sacrifices that opens up our being able to enjoy the fullness of creation as if death were not."
This makes some sense to me. I think this has some similarity to the Moral Influence Theory of the Atonement, which suggests that Jesus's actions inspire us to a better life.
To me, what is intriguing about this Theory of the Atonement is that it emphasizes how Judaism and Christianity were different from the religions of their time. It shows how we were counter-cultural, choosing to show that the One True God is different than what the world had ever really experienced. It basically plots a course of human history that is entirely different than what had been in place prior to that. I think this is fairly consistent with the Narrative of the Scriptures. I think this is a culturally relevant way of talking about the Atonement to someone who has never known Jesus Christ. We all know envy, rivalry, jealousy and striving against one another. We know the violence of the world we live in. The Last Scapegoat Theory of the Atonement gives a way of communicating Jesus Christ as one who provides a different vision for life than envy, jealousy, rivalry, strife and violence. Christ brings peace, hope, love. Instead of working against each other, we can work together in harmony.
I feel like I didn't really do this justice, so do your own reading and feel free to comment and adjust my explanation to more closely depict Girard's thought. I too should do some more study about it.