"I'm sad to leave, but this is a great challenge and I couldn't afford to turn it down," said Sigurdsson. "I would have been silly to say no to this."
Sigurdsson joined Reading in 2005 and had loan spells at Crewe and Shrewsbury before breaking into the first team.
Reading did not reveal how much Hoffenheim had paid for Sigurdsson but did say it was a club record fee received, which means more than the £6.5m Wolves paid for Kevin Doyle in 2009.
Give your reaction to Sigurdsson's departure
Sigurdsson added: "Reading will always have a massive place in my heart and will always be very special to me. I left home to come here when I was 15 and it was hard to say goodbye to everyone.
"But I think Reading have hopefully received good money for me and everything at Reading has been perfect. I feel like I've grown up with the club and the fans have been fantastic to me as well."
Sigurdsson signed a new three-year contract with the Royals in May and had already scored twice in four league games this season, including the opener in Saturday's 2-1 victory away at Leicester.
Sigurdsson was picked out by BBC Sport Football League pundit Steve Claridge as
one of the players to watch
in the Championship this season.
Royals chairman Sir John Madejski said the deal was right for his club: "My support for the football club is unwavering and while I am chairman we will do things the right way.
"We have a well-publicised deficit in our budget now that the parachute payments have ended and we have to take steps to fill this shortfall for the good of the club.
"We wish Gylfi every success in Germany, his rise to prominence has been quite incredible."
Source: BBC Sport
Source: BBC Sport