The overall "goal" of the Twisted. The Babels themselves serves as a "time portal" to the past for the Twisted, and due to the repeated restructuring of spacetime by the deaths of the High Ones, all merge into one giant Grand Babel, which is said to lead to Time Zero. Combine that with the revelation that the Twisted are physical manifestations of the shards of Aya's soul, it seems that the reason why the Twisted kidnap people to make the Babel grow is so that they can return to that specific time and place and reunite temporarily to undo their creation, hence Aya coming back to life in the ending. Couple in with the other point that the only way to do this was for the whole of spacetime to be resturctured, the Twisted actually guided Aya into killing the High Ones, using their own potential threat as an excuse for her to be where the High Ones were.
Remember Hyde? That nice guy that defended Aya from "Boss" and took care of her after she lost her memories? He's actually a High One, created from the remains of Eve's own body and the natural enemy of the Twisted. On top of that, he's trying to go back in time and kill Aya to prevent the creation of the Twisted.
He's not trying to kill Aya. Rather, he's trying to merge with Eve's body after who knows how many loops of the sequence of events to create a new lifeform that can stop the Twisted once and for all, as Aya's fragmented soul meant the Twisted will inevitably appear anyway.