"Everyone who knows me knows how hard I've worked to get here, and I'm proud to be at the best club in the world. I've given my best throughout my career to get here, this was my childhood dream, to step on to this pitch in Madrid's colours," the Belgium international told a news conference on Thursday.
"I received better paid offers but I wanted to be in the best club in the world and be close to my children. Once Madrid's interest was clear I had no doubts that I wanted to be here."
Courtois, 26, signed for Madrid from Chelsea on Wednesday for a reported 35 million euros after winning two Premier League titles as well as the FA Cup and League Cup in England.
Before then, he spent three seasons on loan at Atletico, lifting an unlikely La Liga title in 2014 as well as winning the King's Cup by beating Real in the final at the Santiago Bernabeu.
He also sustained a gut-wrenching defeat to Real in the 2014 Champions League final, when his new captain Sergio Ramos's injury time header denied Atletico glory, leading to Real winning 4-1 in extra-time.
"I have the maximum respect for Atletico, they took me on loan when I was young and I won many things, but it was different because I was on loan," he said.
"Now I've come to the club where I really wanted to be and I want to fight for many years to win trophies."
Courtois's first game for Real could be against Atletico in the UEFA Super Cup on Aug 15 although the fact he has trained little since reaching the semi-finals of the World Cup with Belgium means coach Julen Lopetegui may opt to play Keylor Navas instead.
Courtois missed training with Chelsea in the days before his move to Real was sealed and he thanked the London club for helping him grow as a player but said he could not turn down the chance to play for Real.
"I would like to thank Chelsea's fans for being there for me in the last few years, I always wanted to win the Premier League and I did it twice, but now I have a new chance which I have to take. I also have two children here and that influenced my decision," he added.
"I grew a lot physically in England and now I'm more mature and more experienced. I've improved a lot and I've come here to be the best I can."
(Reporting by Richard Martin, Editing by William Maclean)
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