The midfielder completed his move to the European champions from Bayern Munich just four days after winning the FIFA World Cup™ with Germany, signing a six-year deal believed to be worth around £20m.
He was presented in front of 8,000 fans at the club's Santiago Bernabeu stadium on Thursday evening before taking to the pitch dressed in his new colours and wearing the number eight shirt.
"I know that at this club you are expected to win trophies but I'm the right man for that because I'm used to winning," said Kroos.
The 24-year-old came through the Bayern Munich academy but was entering the final season of his contract with the Bavarian giants. His reluctance to renew with Bayern led Manchester United to pursue his signature, but Kroos said he had no regrets about opting to join Real in the end.
"I know this is the correct decision and I'm very proud to have made it," he said. "I cannot think of anything more wonderful than being here. This is the biggest club in the world.
"In 2013 I won the Champions League and in 2014 Real won it, and I hope we can win it again next year. I hope to be able to return the confidence this club has shown in me."
Real president Florentino Perez lavished praised on his latest recruit, who in addition to the World Cup won that Champions League with Bayern, plus three Bundesliga titles and three German Cups.
"This is a very young player but one with an incredible honours list," said Perez. "We weren't surprised by his performances during the World Cup, they were just confirmation that we had chosen the right player."
Turning to Kroos, he added: "The demands placed on you at Real Madrid are unlimited. You arrive at a club that has just won the European Cup but you have come here to make this club even greater.
"Madridistas always want more and you've come here to make this club even greater, and the fans are already dreaming about winning the 11th European Cup."
Kroos welcomes pressure
In a press conference following his presentation, Kroos reiterated that he was not worried about the pressure he would face at Madrid.
"I like pressure, I have done ever since I started playing," he said. "In Bayern there is also only one option, to win. I'm here to win trophies and that's what stimulates me. I'm going to do everything I can to make sure Madrid continues to have success."
The midfielder also denied he decided to leave Bayern because of a rift with Pep Guardiola.
"Real Madrid are the reason I am here," he said. "I have no problem with Guardiola, quite the opposite. I got on very well with him and have no reason to criticise him. There were no problems with him and I benefited from his style of play."
Instead, Kroos insisted he wanted to leave Bayern, where he made his professional debut in 2007, in search of a new challenge.
"I had spent a lot of time there and won everything. It was the time to leave and the opportunity to come to Madrid, a top level team, arose," he added. "Madrid are the best team in the world and a step above Bayern."
Kroos also revealed he had spoken to Germany team-mate Sami Khedira about Real, but was unable to shed any light on whether the midfielder, who has been linked with a move to Arsenal, would stay at Madrid next season.
"Of course I spoke to him, he told me great things about Madrid and reaffirmed my thoughts," he said. "But what happens in the future is up to him, not to me. It depends on him and Madrid."