The FIFA World Cup star had a rocky season for Spurs last term, despite the team reaching the UEFA Champions League final.
Boss Mauricio Pochettino did not stand in the way of Trippier heading to Atletico, yet he could have used a specialist right-back on Sunday in the north London derby.
Tottenham had to field Davinson Sanchez in that role against Arsenal, with the centre-back out of his comfort zone in the 2-2 draw, while Trippier was helping Atletico to a 3-2 win over Eibar in LaLiga.
"I just feel that Tottenham probably got rid of me at the wrong time," Trippier said. "I just needed that four-week rest for recovery because I was struggling with injuries to come back, fresh, ready.
"It's disappointing. I gave everything for the club and I wanted to stay. I had another couple of years left. But everything happens for a reason.
"I heard rumours about them selling me and it's not nice. At the time, we were going through the Champions League campaign - Manchester City, Ajax away."
Trippier, quoted in several English newspapers, said he spoke with his wife and brother and decided to wait until the close season before approaching Pochettino about his future.
"I tried to speak to the chairman about it and when I spoke to the manager about his plans and I didn't get a yes and I didn't get a no. So you get the impression," he added.
"Me and the manager didn't leave on bad terms, he has done a lot for me and I respect him highly."
Former Burnley player Trippier, who began his career in Manchester City's academy, admitted his form at times last season was shaky.
But working with Atletico boss Diego Simeone looks sure to iron out any tactical weaknesses, with the Argentinian renowned for his strict regimen.
Trippier, 28, said: "For a defender I don't think there's a better place in the world to be."