The 27-year-old signed a six-year deal with the club worth $56m (£33m) after being allowed to leave the Buffalo Bills during free agency.
Byrd told reporters: "Their record and track record speak for themselves. Anytime you can pair up with guys like Drew Brees and Sean Payton with that offense he is running.
"With coach Ryan on the other side and how he turned things around there, that's a combination to win a lot of games. It really sold itself."
Byrd recorded 48 tackles, one sack and four interceptions in the 2013 season.