"I think it's an amazing feeling. I was waiting for this moment for a long time and I am very, very happy to be here," Thiago said.
He added: "When the years are passing, you are trying to win as much as you can, and when you win, you want to win more. I think this club describes what I am as well. I want to achieve all of the goals, win as many trophies as possible."
Liverpool reportedly signed the 29-year-old to a four-year contract worth an initial £20 million. That fee could rise by £5 million if the Reds win the Premier League or Champions League.
Thiago made over 200 appearances after joining Bayern Munich from Barcelona in 2013. He's won the Bundesliga in every season he's spent in Germany and capped off his last campaign at Bayern by winning the Champions League in August.
"I have made the most difficult decision of my sports career," Thiago said in his farewell message to Bayern Munich.
"I will close this chapter on this wonderful club where I have grown as a player for seven years. Triumphs, joy, and also hard times.
"But what I am most proud of is that I arrived in Munich as a young man filled with dreams and now I will leave completely fulfilled with a club, a history, and a culture."
Thiago is Liverpool's second signing of the summer after the Reds added left-back Kostas Tsimikas from Olympiacos.