The veteran wing-back penned a new one-year deal on Thursday.
He said, "It's something I was hoping for and it makes me so happy and motivated. Playing here is special and I'll be able to tell my kids about it for years to come.
“Real Madrid is my home, it's where I have been since I was 16 years old. I've been here for over half my life. It has helped me grow as a person and as a footballer. It's a hugely important part of my life. I feel proud of the fact I've never wavered from my desire to achieve my childhood dream, which was to play for the first team at Real Madrid.
“A footballer is learning all the time. I try and get better with each passing year, setting myself new goals and challenges. That's the basis for maintaining the motivation and hunger to keep winning and improving. That's something they instil in you at the Real Madrid academy.”
On the arrival of Kylian Mbappe, Vazquez told Real Madrid TV: “Kylian is a special player, the kind who was born to play for Real Madrid. We've got an exciting year ahead of us. We're adding an immense talent to what was already here, in Vini Jr., Rodrygo, Jude… All the lads we've got at the top end of the pitch. To have Mbappé here is amazing and we're going to enjoy watching him play this season, and of course, win lots of titles."