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Death told Sam and Dean that God had built Purgatory especially for the Leviathan after Castiel had released and absorbed them. He explained that they were the creatures created before the angels and man. God had locked them away because they were so ambitious, so hungry that they would eliminate anything else in creation. God was afraid that the Leviathan would "spoil the rest of the Petri dish."
Thus far, judging by their plans to turn humanity into a herd for slaughter, Death's assertions have been right. Killing them, as Sam put it, has "been a problem." They've been invincible more or less---save from borax and beheading---but either solution is only temporary. If their heads should be near their bodies, they reattach and go on the offensive once again. Their ability to blend into and become anyone gives them an advantage to infiltrate the world in order to conquer it.
"Reading Is Fundamental" finally answers the question: how does one kill a Leviathan?
We start with Kevin Tran, an ambitious and focused high schooler preparing for college. He is dedicated to his studies---so much so that he times his practices and study sessions. Tran is like a lot of over achieving teenagers: stressed and under pressure. He wants a perfect score on the SAT and to get into Princeton. Later, he informs Dean that he wants to be the first Asian-American President of the United States of America. He is going places if he can stick to his regime.
Unfortunately, as Sam and Dean crack open the hunk of clay that Dick had dug up, he is struck by lightening. It stuns him and leaves him flat on the floor to wake in the morning, his life forever and irrevocably changed. Instead of going to school to complete his test, he drives down the road, driven by the need unlocked. The tablet the Winchesters uncovered pulls on him.
The brothers, as they are setting out to figure out what this tablet is, receive a call from Meg. She has been stationed at Castiel's bedside, watching and waiting for the angel to return to the land of the waking. She tells them if they want answers they'll have to come see for themselves---and their plans change.
Upon arrival, Dean remarks that he is not looking forward to seeing what has become of the fallen angel. He is apprehensive. After all, they had left him behind catatonic and facing the horrors that Sam's Hell memories bring. They are uncertain as to what they will find. Entering the room, they find out quickly that he is nothing like his old self.
Castiel is no longer the serious warrior angel he once had been. He appears more like a child than an angel---going as far as to demand that Dean pull his finger so he can blow the light bulb in the room. Even so, he knows what the tablet is---even if he cannot read it. When Meg tries to look at it, still in his grasp, Dean gets into a heated confrontation with her---which causes Castiel to disappear to the day room.
Dean reluctantly agrees to go and talk to him, and we see Castiel sitting forlorn and alone. He pulls out a board game---Sorry---and invites Dean to play. It is a symbolic gesture---and extremely childlike in nature. He can't make what he did wrong entirely right, but he is making a slight effort. Dean implores him to help them with the Leviathan, to join them in the fight and fix that mistake. Instead, Castiel tells him that he no longer fights. He now "watches the bees."
Meanwhile, Sam and Meg have a confrontation of their own, and she prepares to storm out and take Castiel with her. In his haste, Sam leaves the tablet unprotected on the floor of Castiel's hospital room. Once they settle their differences, they learn that the tablet is gone and Sam chases down the thief---none other than Kevin Tran.
It turns out that while Tran excels at academics he also has another unique gift: he can read the Word of God. The tablet specifically describes the Leviathan---and the piece of the puzzle the Winchesters have been waiting for all this time: how to kill them.
This is why Dick was so frightened, so angered, so anxious in "The Girl with the Dungeons and Dragons Tattoo." This is why he was frantic to get it back before the Winchesters could learn its secrets. Until now, he has been unstoppable, and repossessing this object before its secrets can be learned is imperative.
Before he can send his own after it, however, the angels are one step ahead. They arrive to Castiel's hospital room and attack Sam and Meg to get to Tran and the tablet. It is their duty to take Tran, a prophet, to the appointed desert and learn the Word from him as ascribed in their orders. They are about to take Tran and leave the brothers high and dry when Dean and Castiel return to the room.
These angels are from Castiel's Garrison---from another lifetime. Before they can make their move, Dean slaps his hand on a banishing sigil and zaps all of them away so they can make a getaway. They decide to make the trip to Rufus's cabin, Tran in tow rather he wants to or not.