The 24-year-old completed his switch to the Bernabeu on a six-year deal on Thursday, after making over 200 appearances for Bayern Munich.
Kroos - who joins Real on the back of winning the FIFA World Cup with Germany - has already won three Bundesliga titles, three DFB-Pokals and the Champions League with Bayern.
However, the midfielder is hungry for more success at his new club, and wants to help Real become the first team in the Champions League era to retain the famous trophy.
"I just want to say I'm extremely proud to be here," Kroos said at his official unveiling.
"I could not imagine a more amazing moment than being here with the fans, and I'd like to thank the club for giving me this opportunity to be here.
"They've given me the possibility to play for his incredible club, in my opinion the most important club in the world.
"I want to also thank Bayern Munich for giving me the chance to become a football player at a top club.
"I feel as if I know what it's like at Real Madrid, I know it's necessary to win things and it's a great stimulus for me to try and win things at this club.
"I won the Champions League with Bayern Munich in 2013, Real Madrid won last season. My big objective is to win it again with this club."
Kroos is Real's first signing of the close-season, and could replace his international team-mate Sami Khedira - who is reportedly set to join either Arsenal or Chelsea - in Carlo Ancelotti's midfield.
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