Aiyo, got generation gap, oredi. I feel old.
That line was a classic by De Niro in that movie. He had just bought himself some guns, modified them, and was practising in front of mirror, trying to be tough. He was pretending he was talking to someone, so he says, "You talkin to me? I said, you talkin to me?" and pulls out his gun. he does this several times in the movie.
Anyway, the climax of the movie was De Niro, a taxi driver, goes gun ablazing, killing off some thugs who was holding a girl (played by a young Jodie Foster) for ransom. He gets shot but only wounded -- but saves the day, gets recognition, and gets the respect of a woman (played by Cybil Shepherd) he was trying to woo.
De Niro practices his classic "You Talkin to Me?" in front of the mirror
I said, "You talkin to me?"
Climax of the movie. De Niro saves the girl and kills all the bad guys...bloody nice.