The Unicorn in the Barn by Jacqueline K. Ogburn
A boy, our narrator, discovers a girl putting up "No Trespassing" signs on his family's land -- he tells her that her mother had not bought that part. Then he goes up into a treehouse and sees a unicorn, limping.
Following it takes him onto the land they had sold, where the new owner was a vet, and gets him caught even though he was just trying to see if the unicorn was all right. It gets him a job working there after some talk.
The details of his life come out as he works with a half-invisible cat and other magical creatures: his grandmother is in the nursing home, and his mother had died when he was two, and he lives with his father and older brother. He learns about life and death and the limits of magic, and other complications, as the story weaves on.