Eddie Ray Routh is accused of killing the former Navy SEAL and Mr Kyle’s friend Chad Littlefield at a rural shooting range in February 2013. Before resting, prosecutors in Stephenville, Texas, played a recorded phone call between Routh and a reporter from The New Yorker magazine in which Routh talks about the events, The Dallas Morning News reported.
“I had to take care of business. I took care of business, and then I got in the truck and left,” Routh said in the call.
Routh also said he was annoyed that Mr Littlefield was not shooting.
“Are you gonna shoot? Are you gonna shoot? It’s a shooting sport. You shoot,” Routh said in the phone call. “That’s what got me all riled up.”
Mr Kyle served four tours in Iraq and made more than 300 kills as a sniper for SEAL Team 3, according to his own count. He earned two Silver Stars for valour.
After leaving the military, he volunteered with veterans facing mental health problems, often taking them shooting.
He wrote a best-selling memoir that was the basis for the box office smash American Sniper.
Routh, whose lawyers are mounting an insanity defence, has pleaded not guilty.
Family members say Routh suffered from post-traumatic stress disorder from serving in Iraq and in Haiti after the devastating 2010 earthquake.
Defence lawyers say Routh, who was taking anti-psychotic medication, was insane when Mr Kyle and Mr Littlefield took the former Marine to the shooting range to provide support and camaraderie.
Routh, his lawyers say, believed that the two men planned to kill him.
Prosecutors say Routh was a troubled drug user but that he knew right from wrong.