Rufus wanted to kill himself because his father beat him.
Rufus’s father can sometimes be uncaring and cruel. Rufus respects some things about him, such as his apparent honesty, but is afraid of him and hates him. His father beats him harshly, and does not seem to care whether he lives or dies.
The boy had his back to me and hadn’t noticed me yet. He held a stick of wood in one hand and the end of the stick was charred and smoking. (p. 19)
In anger, Rufus uses a stick to set fire to the drapes in his bedroom in the middle of the night. This action puts his life in danger and pulls Dana from the future to rescue him.
Dana grabs the curtains and throws them out the window, only later realizing she could have just thrown them in the fireplace. The danger is gone, but she is there in the past.
Dana is angry at Rufus at first. She blames him for pulling her into the past.
I stepped over him, took the stick from his hand, and threw it into the fireplace. "Someone should use one like that on you," I said, "before you burn the house down." (p. 21)
Angry, Rufus tells her not to lay a hand on him. Even before he says this, Dana regrets having threatened him. She tells him softly that he is going to be in trouble anyway when his father finds out about the drapes. She understands she is going to need his help, and should not antagonize him.
Rufus shows Dana welts on his back, and she sees scars from where he has been beaten many times before. He also confesses that he once got mad because his father did not give him a horse, so he burnt down the stable.
Rufus did not try to kill himself to be saved by Dana, because until this incident he did not realize that she would be pulled through time when his life was in danger.