He asked me why I want him here

Guys... I need advice. My ex boyfriend's son is staying with me. He's 19 and his dad constantly beat the shit out if him. He definitely has a mouth in him, but I understand why now... Because of abuse. His dad has pulled a gun on him several times, one of them I was there for. I started checking up on him a while after me and his dad broke up and the moment I decided to get him was when his dad pulled a gun on him again and fucked him in the mouth with a gun. He's been staying here over a week. When I first got him he said he had a problem and needed to get something from the store before he could sleep at my house. I said what was it and he said nevermind since he would only stay a few days. He's trying to redo his senior year but has been falling behind because he refuses to sleep at night. He's been staying in my guest room. He set an alarm for 10 minutes to take a 10 minute nap and I turned it off so could get some sleep. He slept for 13 hours and woke up freaking out. I said he deserved the sleep but noticed he had an... Accident on my couch. He yelled at me not to look at him so I didn't. I had depends sent to my place and said nothing. This was what he was afraid to tell me. We talked and I talked to him about his nightmares. He talked about his mom.... Said she was on drugs and neglected him. Never took care of him and he walked in on her overdosed.... He asked me a question... "Why do you want me here. I'm your ex boyfriend's adult son. You guys didn't even date that long. I don't understand why you even want me here". I didn't have an answer. Everyday he packs his stuff to find somewhere to go and everyday I tell him to unpack it and he's not going back to his dad or be homeless. He should finish school and become better than his parents. How do I show him I want him here?

Edit: For the therapist suggestion I was wondering if he would qualify for sooner care. He has no income and is still in high school.