Bridcutt joined Forest from Championship rivals Leeds earlier this week for a transfer fee reported by The Nottingham Post to be worth £1 million.
The 28-year-old defensive midfielder was struggling for playing time at the Whites this season under new manager Thomas Christiansen.
The former Sunderland man has admitted his surprise at falling down the pecking order at Elland Road this campaign.
When asked if he was surprised, Bridcutt told Sky Sports: "Definitely. You never really see it coming.
“But as I say it's their loss and Forest's gain, and for me it was a difficult point and there was no reason behind it.
"It was a new manager, and he obviously had his select of players and I wasn't one of them. For me now it's just to prove everyone wrong.”
Explaining his decision to join Forest, Bridcutt said: "The big point of me moving was to do with the manager.
“I've seen him work previously, the style of football his teams have played, and I think it suits me massively. Everyone who I've spoken to that has worked with him have always said positive things to say.”
Bridcutt could make his debut for Forest against his former club Leeds in the Championship game at the City Ground on Saturday evening.
Forest manager Mark Warburton has said that he is confident that the Scotland international will be motivated to do well against the West Yorkshire outfit.
"He is a professional player who has joined a new club," Warburton told Forest’s official website about Bridcutt. "I am sure that whoever we were playing against he would be more than motivated.
“He wants to impress his peers, he wants to impress the fans and he wants to show himself. He has missed some football but his fitness levels are good, we are happy with where he is at and I have no doubts he will be ready if he starts or comes off the bench.”
Both Forest and Leeds have made good starts to the 2017-18 Championship campaign, and will be aiming to finish in the top six of the league this season.