The goalkeeper joins from SC Freiburg on a four-year contract.
"I'm very excited," said Flekken. "It's a whole new chapter for me and the next big step in my career.
“I'm really looking forward to my time here at Brentford and it's really lovely to be here.
“If you had told me seven years ago when I was in the third division of Germany that, one day, I would be playing in the Premier League, I would have probably knocked you down!
"But it's a dream come true and my next big step. And I hope we can take the next steps as a squad and as a club as well."
He added: "I've spoken with [Thomas Frank] a few times and even exchanged some text messages with him recently.
“He's a really, really good guy and a great coach, as I'm hearing from everywhere, so I'm looking forward to meeting him more closely.
“He told me that I would be a perfect fit for the club over here and that I would fit in really well. I'm really looking forward to playing my part here."