The 24-year-old has admitted that his initial plan was to stay at the Allianz Arena, but he is now looking forward to the challenge of playing for the European champions
The Germany international was keen to stay in Bavaria, but he eventually opted to join the European champions for a reported €25 million (£20m) after failing to agree terms on a contract extension with Pep Guardiola's side.
"I first started thinking about a transfer when talks with Bayern were unsuccessful. Bayern were always my priority and I didn't talk to anybody else," Kroos told the official DFB website.
"I then decided to go for a new challenge after we failed to reach an agreement with Bayern and a number of clubs showed an interest in signing me. Winning the World Cup settled it."
The versatile midfielder dismissed rumours that he left the Allianz Arena because he did not feel appreciated by the Bundesliga titleholders.
"I didn't leave Bayern because I didn't feel valued," he insisted. "I played for one of the biggest clubs in the world and was a starter under several coaches.
"I always played in the major competitions and Bayern always looked after me. [Coach] Pep Guardiola and [sporting director] Matthias Sammer were important to me and the fans always supported me as well.
"But Real Madrid are a huge challenge. People all over the world know Madrid.
"Thousands of people attended my presentation at the Santiago Bernabeu and club president Florentino Perez was interrupted a number of times by cheering fans. I have never seen anything like it.
"This club is just the right step for me."
The 24-year-old faces stiff competition for a place in midfield at Madrid, but he has little doubt that he will become a regular at his new club.
"You have to deal with that as a footballer," he reasoned. "I am looking forward to playing alongside guys like Cristiano Ronaldo, Sergio Ramos and Sami Khedira.
"No team has ever won back-to-back Champions League and that's exactly why Madrid have signed some new players.
"It can only help to bring in new players who want to prove themselves and challenge the established players."