It was done in typical "Hollywood" fashion - with permits, lots of security and stunt people. You don't get a long, complex and really terrific scene like that without a lot of planning and a lot of takes.
From Wikipedia:
Car chase
The film is often cited as containing one of the greatest car chase sequences in movie history.[9] The chase involves Popeye commandeering a civilian's car (a 1971 Pontiac LeMans) and then frantically chasing an elevated train, on which a hitman is trying to escape. The scene was filmed in Bensonhurst, Brooklyn roughly running under the BMT West End Line (currently; the D train, then the B train) which runs on an elevated track above Stillwell Avenue, 86th Street and New Utrecht Avenue in Brooklyn, with the chase ending just north of the 62nd Street station after the train crashed into another train up ahead. The conductor played by Bob Morrone and train operator played by William Coke, aboard the hijacked train were both actual NYC Transit Authority employees.[unreliable source] Friedkin's plan included fast driving coupled with five specific stunts:
Doyle is sideswiped by a car in an intersection
Doyle's car is clipped by a truck with a Drive Carefully bumper sticker.
Doyle narrowly misses a woman with a baby stroller and crashes into a pile of garbage.
Doyle's vision is blocked by a tractor trailer which forces him into a steel fence.
Doyle must go against traffic to get back on a parallel path with the train. Intercut with these car scenes underneath the elevated train is additional footage (shots facing the car, not from the driver's perspective) that was shot in Bushwick, Brooklyn, particularly when Doyle misses a moving truck and slams into a steel fence.
The most famous shot of the chase is made from a front bumper mount and shows a low-angle point of view shot of the streets racing by. This was the last shot made in the film and was, according to Friedkin, needed to increase the speed of the chase after a rough cut of the scene proved less impressive than he hoped. While Friedkin contends the front-bumper shot is made at speeds of "up to 90mph," director of photography Owen Roizman, wrote in American Cinematographer magazine in 1972 that the camera was undercranked to 18 frames per second to enhance the sense of speed. Roizman's contention is borne out when you see a car at a red light whose muffler is pumping smoke at an accelerated rate. Other shots involved stunt drivers who were supposed to barely miss hitting the speeding car, but due to errors in timing accidental collisions occurred and were left in the final film. Friedkin said that he used Santana's song "Black Magic Woman" during editing to help shape the chase sequence; though the song does not appear in the film, "it [the chase scene] did have a sort of pre-ordained rhythm to it that came from the music."
The scene concludes with Doyle confronting Nicoli the hitman at the stairs leading to the subway and shooting him as he tries to run back up them. Many of the police officers acting as advisers for the film objected to the scene on the grounds that shooting a suspect in the back was simply murder, not self-defense, but director Friedkin stood by it, stating that he was "secure in my conviction that that's exactly what Eddie Egan (the model for Doyle) would have done and Eddie was on the set while all of this was being shot."
What he is probably talking about in the video is the "front bumper" extra footage shot after the rough cut.
