While the Tudor City scene does nothing to advance the movie's plot, it chillingly illustrates Bickle's precarious state of mind, fueled by the degenerates he encounters on NYC's mean streets.
As the scene begins, Bickle picks up a fare in front of McAnn's Bar and they proceed to Tudor City.
The cab drives up 41st Street past the South Park...
...and stops in the cul-de-sac adjacent to Tudor Tower.
(The corner store, today Conran's Bike Shop, was then a Thomas Cook travel agency.)
It's quickly apparent that the passenger (played by a jittery Martin Scorsese) is completely deranged. He tells Bickle that his wife's having an affair with a man in a Tudor Tower apartment.
... the passenger's wife, languidly smoking a cigarette and gazing out at the lights of the U.N.
The passenger spells out his plan for revenge in a long, scatological monologue. "Did you ever see what a .45 Magnum pistol can do to a woman's face?" he asks. The taxi driver listens numbly. There's an abrupt cut to the next scene.
See the full, R-rated sequence here.