The Senegalese striker has left Liverpool to join the German giants in a £35 million deal and is looking ahead to a new challenge after six years with the Reds.
Mane said: "I'm very happy to finally be with FC Bayern in Munich. We had a lot of talks and I felt the great interest of this big club from the start.
"So there was no doubt in my mind right from the start: This is the right time for this challenge. I want to achieve a lot with this club, also internationally.
"During my time in Salzburg I followed many FC Bayern games - I like this club very much!"
Mane had "no doubts" about leaving Liverpool but stressed the importance of departing the Premier League on good terms.
He explained: "I sent a long text to say goodbye to everybody, which is normal. They were sad, as I was, but it is part of life. We just have to accept it.
"That was the plan from the beginning - to leave Liverpool in a very good way. It was fantastic in the club, in the dressing room.
"I am friendly with everybody, the coaching staff and even the kitchen staff, the ladies in the kitchen! So, for me leaving in a good way is so, so important - and [for] the fans as well."
Mane won the Champions League as a Liverpool player in 2019 and is now hoping to replicate the achievement at Bayern.
He said: "Every kid wants to win the Champions League, every football player in the world. I’m with a very, very competitive team now.
"But it’s still a bit early to talk about the Champions League final, we haven’t even played a game together yet. Nevertheless we will give everything to get into the final."