Atleti parted with £21.7m to sign the 28-year-old, according to Sky Sports.
"It is an amazing feeling. I've always wanted to play abroad and what a place to play in Madrid," Trippier told Atletico's website.
"It is a fantastic team with an amazing manager and I'm just looking forward to getting on with it now and meeting the team.
"They've been to some major finals, they've done so well in La Liga over the years. They've got a quality team and an unbelievable manager.
"It is something I can't wait to do, to learn off the manager and help the team to be a success in the future.
"The manager, you can see how close he is over the years with his squad, I can't wait to be part of it."