Director
Martin Scorsese
An Intense Dude
The great movie critic Roger Ebert said that director Martin Scorsese was "one of the most intense people I'd ever known" (source). (Scorsese's still very much alive and active, if the past tense of that observation confuses you; Ebert, however, passed away). That intensity suffuses every frame of Taxi Driver: It's a movie about a really intense dude made by another really intense dude.
The director Brian DePalma (who did the original Carrie) introduced Scorsese to Paul Schrader, hooking him into the Taxi Driver project, which took about four years to finally get made and appear in theaters. Unlike a lot of other people who looked at the script, Scorsese was way into it—and it inspired him, gave him more ideas. For one thing, he wanted to hit a dreamlike tone in making the movie, and also wanted to work in references to his own Catholic upbringing (e.g. the Mass-like way in which Travis burns candles and lights cans of shoe polish), since he saw Bickle as a kind of twisted parody of a saint, attempting to cleanse his body and mind.
He cited Alfred Hitchcock's The Wrong Man, Francesco Rosi's Salvatore Giuliano, and Jack Hazan's The Bigger Splash as inspirations for the movie's camerawork (source).
Macho Men
Thematically, Taxi Driver has a lot in common with Scorsese's other movies, which tend to explore the darker side of American masculinity. This is true of Raging Bull, Goodfellas, The Wolf of Wall Street, and plenty of others besides.
Along with Taxi Driver, the three movies mentioned all take a male character who has qualities of strength and energy—but isn't exactly a sensitive poet-philosopher—and then throw him into a situation in which those qualities gradually go out of control and become totally destructive.
Because Taxi Driver critiques this kind of aggressiveness, Scorsese says the movie is
"Feminist. Because it takes macho to its logical conclusion. The better man is the man who can kill you. This one shows that kind of thinking, shows the kinds of problems some men have, bouncing back and forth between the goddesses and whores. The whole movie is based, visually, on one shot where the guy is being turned down on the telephone by the girl, and the camera actually pans away from him. It's too painful to see that rejection." (Source)
Whether you find the movie feminist or not, you'll probably agree that Scorsese does a fantastic job penetrating the mind of a lonely and highly unstable man. He makes sure that the film's visuals perfectly capture the isolation and the descent into madness presented by Schrader's script.