The England international completed the move while representing his country at the FIFA World Cup, ending a memorable two-year spell at the City Ground.
Anderson made 92 appearances for Forest, playing a key role as the club returned to European competition for the first time in 30 years.
“Forest family, Firstly I am so grateful for the love and support you have shown me over the last 2 years especially towards the end of this season. From the minute I walked through the door, I fell in love with this club,” he wrote on social media.
“I want to thank Mr Marinakas, the staff and all players for welcoming me and making my time here so enjoyable, this club gave me the opportunity to grow as a player and a person.
“Bringing the club European football again after 29 years is what all of you fans deserved and to be a helping hand in that is so special.
“Nottingham Forest will always have a special place in my heart. Thank you for all the unforgettable memories.”