What's the meaning of the expressions "break my chops" and "take it on the arches"?
Peter J is correct as to the ultimate meaning, but the specifics are what make these expressions interesting.
To "take it on the arches" means to leave, specifically
To depart, especially on foot.
Wiktionary
Wiktionary further explains that the expression is "a reference to fallen arches that result from too much walking." It's an insult, suggesting that the person addressed is probably too much of a deadbeat to afford any other mode of transportation.
"Break my chops" is the more interesting of the two, since it is an admixture of two expressions: break my balls and bust my chops. Here it really favors the "break my balls" side, which means;
- vulgar slang To tease one; to give one a hard time.
TFD Online
To "bust (one's) chops" usually means
- (acting upon oneself) To exert a significant amount of energy or work very hard to do, accomplish, or complete something.
TFD Online
Clearly the taxi dispatcher is not complaining about Travis Bickle exerting a significant amount of energy to accomplish something. What he's doing is mixing his expressions, which he does wonderfully in the way people actually speak in real life much of the time, and which is also what makes the dialogue in this film seem so real.