After defeating the first boss, Gyoubu Oniwa at the end of the Ashina Outskirts, you can head up the stairs to reach another building. You’ll find the Tengu of Ashina there and he’ll ask you to assist him in killing some Rats, handing you the Rat Description item in the process.
After Killing the Rats
From the Ashina Castle Gate Idol, head up the path and then over to the right where you’ll find the Rats he’s seeking. After putting them down, return to the Tengu of Ashina to obtain the Ashina Esoteric Text and unlock the Ashina Arts tree.
During the Second Visit to Ashina Castle
After you return to Ashina Castle after retrieving the items from the Sunken Valley and Ashina Depths, you’ll find the Tengu of Ashina at the Great Serpent Shrine Idol.
Speak to him and he’ll ask if you’ve mastered any techniques yet, this refers to the final skill in any of the current trees you have unlocked. Once done, return to him and he’ll hand over the Mushin Esoteric Text, unlocking the Mushin Arts tree.