The tree library is the tree located on the Temple island that had the original sacred Jedi texts while Luke Skywalker was exiled on Ahch-To. The library was destroyed by a lightning storm summoned by Yoda's Force spirit, but Rey took the sacred texts with her before leaving the island and they currently remain in her possession.
The tree library is an uneti tree, a species of tree that grows on Ahch-To. It has a hollowed out tree trunk with a reading chamber inside, which has a bookshelf containing a collection of the original sacred texts of the Jedi religion. The texts were assembled by the Jedi Master Luke Skywalker and placed in the tree during the period he exiled himself on the island.
Rey arrives the Temple island on Ahch-To to find the Jedi Master Luke Skywalker and asks him to fight for the Resistance. However, Luke refuses to leave Ahch-To, wishing to die exiled from the rest of the galaxy, putting an end to the Jedi. Rey follows him around the island trying to convince him to help the Resistance, but then she hears whispering voices.
The voices lead Rey to an uneti tree, and inside she sees that they are coming from a collection of books on a shelf. As she approaches to touch the books, Luke enters the library and asks her who she is. Rey replies to him she is no one from nowhere (Jakku), but Luke still seems to be intrigued. While examining the place, Rey admits to Luke that she has seen the library and the Temple island in her dreams. She tells Luke that she needs help with her newfound Force powers, but Luke refuses to train a new generation of Jedi again.
Afterwards, at the night Rey confronts Luke about his attempt to murder his nephew, she leaves the island, and Luke decides to destroy the tree library. But as he approaches the tree, Luke is surprised to see Yoda's Force spirit. Yoda greets his former apprentice, and when he sees that Luke is hesitating to burn the library, he uses the Force to summon a lightning bolt that sets the tree ablaze.
When he sees the tree on fire, Luke regrets his previous decision to destroy it and screams to Yoda to save the sacred Jedi texts. But Yoda only smiles and says that there was nothing inside the library that Rey doesn't already have. The two Jedi Masters then talk about failure and Yoda advises Luke to learn from his past mistakes with young Ben Solo to become a better teacher as they both watch the tree library burn. Unbeknownst to Luke, Rey had taken the sacred Jedi texts from the tree library and stored them inside the Millennium Falcon before leaving the island.