The Scotland star has departed Chelsea to get more game time.
Gilmour, who starred for the national team against England at Euro 2020, has made 22 first-team appearances across the past two seasons.
He was linked with a return to Rangers but is staying in the Premier League.
"I spoke to the manager here [Daniel Farke],” Gilmour told Norwich City's website. “He called me and we had a nice talk about how he wants to play and how he sees me fitting in. Everything he said was what I want to do on the pitch, get the ball down and play nice football. He spoke really highly of his team and I wanted to come join here and help.
"He said it’s a very good bunch of boys, which I can’t wait to see, and the way we play football, the style of getting the ball down and playing, is not like many other teams which punt it up the pitch. We stick to our philosophies. He was speaking very highly of the training they do and the coaching staff and the facilities and that’s what made up my mind.”
Gilmour also revealed that a dinner with Norwich owner Delia Smith helped seal the deal.
He continued: "She just loves football, doesn’t she, and she loves Norwich City.
"She was really happy for me to be here. She was talking about how good the Euros was and how she’s still watching it.
"She’s a lovely woman and I look forward to seeing her again.
"I’m very excited to be at Norwich. I can’t wait for the season ahead.
"I’ve been looking forward to coming here, playing as much as possible and helping Norwich City compete in the Premier League.
"It was a massive experience to go over there to Porto and win the Champions League with Chelsea.
"It’s a night I’ll remember forever. That’s in the past now, I’m here at Norwich and ready to do big things."
Gilmour left Rangers in 2017 to move to Chelsea at the age of 16.
He won the club’s academy player of the year in 2020, by which time he had progressed to the first-team under then boss Frank Lampard.
It was reported that as part of the deal which took him to Stamford Bridge for an initial £500,000, Rangers are set to earn another £1million due to his progression.
He was an unused sub for Chelsea’s Champions League win over Manchester City in May,
“We are delighted to bring Billy in because we’ve lost some players in that area,” Norwich manager Farke said. “We’re looking to improve our options, particularly in the centre of the park, and for that we’re happy to bring Billy in.
“We got the feeling that he can help us a lot with his ability to link the play, his technical qualities and also his work against the ball. From the football point of view, but also from his character he is the perfect fit to our squad.”