The Bundesliga side managed to beat a number of Premier League clubs to the signature of the 19-year-old.
Burke played 30 times for Nottingham Forest, scoring five goals, with four of them coming this season.
The Scotland international released the following statement on his Twitter account.
"When I found out that Forest had agreed a deal to sell me it was a surprise, but I understand that the transfer fee received should help stabilise the club and allow Forest to sign at least two or three quality players who can help push the team for promotion.
"As for my new club, RB Leipzig, all I can say is they wanted me and expressed a real hunger for me to join them and hopefully enhance their team".