What proof leads Montag to the realization that "Beatty wanted to die" in Ray Bradbury's Fahrenheit 451?
After killing Captain Beatty, Montag reflects on what happened and has an epiphany. He is convinced that Beatty wanted to die and he comes to this realisation because he knows that Beatty could have saved himself, should he have wished it. Instead, Beatty made no real attempt to fight Montag off:
He had just stood there, not really trying to save himself, just stood there, joking, needling.
In fact, Montag suggests that Beatty deliberately provoked him because he wanted to be murdered:
You go on yelling at people and making fun of them until you get them mad, and then....
This epiphany reflects Montag's views about the wider society: that book-burning and censorship benefits nobody and instead makes people intensely miserable. In this respect, Beatty is just like Montag's wife, Mildred, who takes an overdose in Part One of the novel.
This epiphany is significant because it suggests that Montag is not a criminal and certainly does not view himself in such a way. To some extent, he has provided a service to Beatty by putting an end to his miserable life.
After Guy Montag kills Beatty at Montag's house, he is running away (or trying to since his leg is all messed up because of the hound. He falls down and when he does, he realizes that Beatty wanted to die.
What he gives as evidence is Beatty's behavior. He says that only someone who wanted to die would taunt someone who had a weapon like that (like the flamethrower that Montag had). Someone who didn't want to die would have been leaving Montag alone so that they could stay alive instead of taunting him.