Graff is explaining to Ender the philosophy behind everything they are doing. Although Ender does not know it at the time, this is the same reasoning that the adults will use to manipulate the children time and time again. Ender objects to this idea, because he believes that people are more than just tools, but nevertheless it is the pervading ideology of the I.F. throughout the book. This philosophy justifies doing terrible things in the name of humanity, and it also means that individuals will have to make awful sacrifices for their species.
"In the moment when I truly understand my enemy, understand him well enough to defeat him, then in that very moment I also love him. I think it's impossible to really understand somebody, what they want, what they believe, and not love them the way they love themselves."
Ender is telling Valentine why he hates himself. He is able to understand his enemies better than anyone else, but once he understands them he destroys them. With such tremendous empathy, even in coming to understand his worst enemies Ender loves them. This means that when he crushes them he is hurting himself in the process. Ender does not want to have enemies, so that he will not be forced to hurt anybody. He will love even those who seem to be his most bitter enemies when he properly understands them. But in the situations Ender has been in he has no choice other than destroying those enemies. At the end of the book, when Ender comes to truly understand the buggers, he is able to try to help them. He has already done them great harm, but now he can be happy because he has a chance to undo what he did to them.
"So the whole war is because we can't talk to each other."
"If the other fellow can't tell you his story, you can never be sure he isn't trying to kill you."
"What if we just left them alone?"
"Ender, we didn't go to them first, they came to us. If they were going to leave us alone, they could have done it a hundred years ago, before the First Invasion."
"Maybe they didn't know we were intelligent life. Maybe—"
This conversation occurs when Graff tells Ender his theory of why they are at war with the buggers. Graff tells Ender that since the buggers communicate through thought, they probably cannot understand that humans are thinking beings. Ender therefore wants to know why this cannot be remedied. Graff p