"I've always been interested on a personal level in helping our vets. I've been a part of USO on tours for over eight years," Cooper tells PEOPLE in this week's cover story on Kyle and his widow, Taya.
"I had an appreciation for veterans before this, but what I definitely did not know is the toll that it takes on the family," the Best Actor Oscar nominee says.
Cooper and Taya Kyle have hosted screenings of the movie and visited hospitals, meeting veterans from all over the country.
"People were willing to express themselves in a format that they would never do normally," Cooper says. "But because they saw Chris's story, they were willing to say, 'Thank you for putting a guy I can relate to up there and have it be something right away that I know is accurate.' "
Says Cooper: "It's been life-changing."
Cooper says he is "gratified" that the film, which shows the Kyles struggle during Chris's deployments and after he left the military, is resonating with veterans.
"People who see it can relate to it and go, 'Well, if he goes through that, then maybe it's okay that I'm going through that," he says. "'If his family had to deal with that stuff, then maybe I am not alone.' That's a lot of what Chris was doing when he got home. He was telling his story. He got his life together, and a lot of that is when he found a purpose in being able to help vets."
Cooper "gets it," Taya says: "He went above and beyond just acting in a role. He got into the mind and heart of Chris. You can't do that and not come out with a special appreciation for guys who lost a limb for this."
While Cooper says he is "lucky" to have been nominated for an Oscar for the past three years, this year's Best Actor nod for playing Kyle has extra meaning.
"I would be lying if I said I didn't get emotional when I heard my name and saw Chris's picture up there," he says. "I thought, 'Thank God,' because I felt the role merited that. I just hoped I did a good enough job for the Academy to recognize that his story is worthy to be a part of these other people's stories."
For more from Bradley Cooper – and Taya Kyle's emotional interview about her life with Chris – pick up the latest issue of PEOPLE, on newsstands Friday