The false claim is that anyone in attendance mistook Kennedy's statement to mean anything other than "I am a person from Berlin." It is equally false to claim that Kennedy misspoke or made a gaffe in this speech. Only later was this double meaning of the word 'Berliner' pointed out by comedians.
The sad part is that there are people who insist that one can hear audience members laughing at that moment in recordings of the speech. Nope, just cheering.