The midfielder has signed a four-year deal after the Royals agreed an undisclosed fee with the German club.
Williams, 24, joins a squad already strengthened by the arrivals of Wayne Bridge and Royston Drenthe ahead of the Championship season.
He told the club's official website: "I am very excited to be here, I am really looking forward to the season and to try to get the club back to the Premier League.
"I spoke to the manager Nigel Adkins when I did my medicine check and he is a really great, nice guy. I'm looking forward to working with him.
"I would like to say to the Reading fans I will give 100% every game, I am proud to wear this jersey and will do everything to help this team back to the Premier League."
Director of football Nick Hammond said: "Danny is a versatile player, he possesses great energy and already has great experience in one of the top leagues in Europe.
"He is an international player and at 24 years of age still has plenty of development ahead of him, so we will be delighted to welcome him to the club."