GA is a new way of thinking about human origins, and the anthropology of the human being. The anthropological discourse of the day always serves as an implicit structure in our conversations about politics, religion, economics, and our social lives. If we misunderstand our own anthropology, we will be forever mystified by our inability to make progress on some of these difficult questions.
The original insights that ultimately gave rise to Generative Anthropology come from the French Literary Theorist Rene Girard. Girard revealed the "mimetic nature" of human beings and higher primates in general. If you're unfamiliar with Girard's work, it would be good to familiarize yourself with elements of his theory although it is not strictly necessary since most of the GA theorists explicate the essential Girardian insights in their work.
Where GA expands upon Girard is in its revision of the origin of the human. This process of "homonization" or the transition from a last common hominid ancestor into the being that we call the human is the cornerstone hypothesis around which the rest of the discipline of GA revolves.
The Originary Hypothesis of Generative Anthropology is the idea that increasingly mimetic hominids eventually reached a situation in which their primate dominance structures were unable to contain the new violent possibilities enabled by their increasingly mimetic natures. An alpha hominid can dominate a beta hominid but an alpha hominid cannot dominate an entire pack of hominids. The Originary Scene is a hypothetical scene of origin of the human in which a group of mimetic hominids surround an object (for example, an animal caracass) and because of their mimetic natures they all attempt to appropriate the carcass at one time. As they approach it one of the hominids hesitates, and "aborts his gesture of appropriation" likely by pointing at the object in question. This hesitation and the gesture of the pointing is then mimicked by other hominids who realize the danger present if all approach the object at the center at one time.
This gesture, recognized by all as "more" than just a gesture, is the first shared meaning in the universe. The sign has a signifier and a signified and represents not itself but "picks out" another object in the universe. This is the first word and the beginning of language. Meaning now exists, and the human begins.
This origin story is powerful because it gives us a scene in which any human ability: our ability to do science, to be ironic, to love, to think, to dominate, etc can then be carefully explained first by reference to this scene of origin. The fundamental elements of the human are present on the scene. Desire, Resentment, Envy, and even Love. From these fundamental elements we can then construct a new and more accurate anthropology of the human being. What follows from this new construction, this new vocabulary, and this new way of thinking is nothing short of a miracle.