Much of this is based on the philosophy of Pierre Tielhard de Chardin, which posits that (very loosely speaking) humanity as a whole can evolve towards becoming one with God (in essence, humanity is God in the making).
In the books, the TechnoCore is working on a parallel project, creating an Ultimate Intelligence (UI) which can use all available knowledge to predict the future. In essence, they want to create a god-like intelligence analogous to what humanity will evolve into.
Far, far in the future, both goals are achieved. The UI and what the Core refers to as "the human UI" are at war, with the the stakes being the future fate of humanity. The human UI, however, produces a stalemate by refusing to fight, and flees backwards in time instead.
Back in the present, the TechnoCore learns from their UI (who can also communicate back in time--don't ask how) that the human UI is in their time, but his location and identity are unknown.
In order to find the human UI, though, they build a "trap":
The Tree of Pain will call him\\
The Shrike will call him\\
to lure the human UI (who, among other things, is the personification of empathy) out of hiding in an attempt to help those trapped on the Tree of Pain. Once the human UI has revealed himself,
The true UI will destroy him\\
that is, the UI will be able to resume the fight and (as the TechnoCore believes) defeat the human UI.