The 32-year-old midfielder signed a two-year deal on Wednesday that will keep him at the Bernabeu until at least 2016.
Alonso, who was due to become a free agent in the summer, said in a press conference: "I have always known that Real Madrid is the best place I can be.
"In Madrid, my wife and I have found ourselves very well. We had the total conviction to stay here."
Alonso suffered an injury to his eardrum in December and is yet to return to full fitness, and said this delayed him in signing the new deal.
"It's been a process (to sign the new deal). There has always been communication with the club," he said.
"After the operation, I wanted to see how I was, what level I could give and it was a huge decision.
"I did not want to take it before I knew if I would be okay but the club gave me every help.
"When I was convinced, everything has been very straightforward. I am very happy and the club says that they are happy too."
Alonso added: "I am here to express my happiness and commitment to Real Madrid for two more seasons. I feel at home, I took five very nice seasons and I hope the next two are as well.
"I thank the club and the fans for their support."