The Germany midfielder joined European champions Real on a six-year deal for an undisclosed fee last week after months of speculation over his future.
Kroos, 24, won three Bundesliga titles, three DFB-Pokals and the UEFA Champions League during his illustrious spell at Bayern.
He then played an integral role in Germany's World Cup triumph in Brazil before opting to join Carlo Ancelotti's side, but said he only left the Allianz Arena after failing to agree a new deal as he entered the final year of his contract.
He told the German Football Association's website: "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.
"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."
Kroos has no hard feelings with his former employers and is relishing the opportunity to showcase his talents at Real.
"I didn't leave Bayern because I didn't feel valued." he added. "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. Pep Guardiola (Bayern coach) and Matthias Sammer (Bayern sporting director) 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."
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