The scene in which Travis is talking to himself was completely created by Robert De Niro(listen to the speech); the screenplay simply says: “Travis looks in the mirror”. De Niro repeats himself, because Martin Scorsese was listening to the takes on his headphones, and with the background noise, Scorsese did not know if De Niro could be heard clearly. The line itself derives from the classic 1953 western Shane.
I’m faster than you, you fuckin’ son of a…I saw you comin’, you fuck, shit-heel. I’m standin’ here. You make the move. You make the move. It’s your move. Don’t try it, you fucker. You talkin’ to me? You talkin’ to me? You talkin’ to me? Well, who the hell else are you talkin’ to? You talkin’ to me? Well, I’m the only one here. Who the fuck do you think you’re talkin’ to? Oh yeah? Huh? OK. Huh?