The 28-year-old England international left back joined City for an undisclosed fee and signed a four-and-a-half-year deal at Eastlands on Saturday morning.
"I'm just excited to be here and when I spoke to the manager yesterday he told me of the exciting times ahead at the club," he told the club's official website.
"It's an exciting time to be at the club with what's happening here and I'm looking forward to putting the shirt on regularly and getting out there to show everyone what I can do."
Bridge is lacking match fitness due to Ashley Cole's fine form at Stamford Bridge, but the England international is hopeful of getting back in the groove in no time.
"I'm just looking forward to getting out and playing some regular football," he continued.
"I haven't played that much, so I'm aiming to play on a regular basis, get my fitness up to scratch and hopefully help the team climb the table and achieve big things."