。
I promised her I'd be home by 9.
Literal
I [topic-は] 9 o'clock [by-までに] return [quotative-という] promise [object-を] her [to-に] did.
という here is a quotative complementizer — links the embedded clause '9時までに帰る' to the noun 約束 ('promise'). This [clause]+という+[noun] structure is how Japanese forms 'noun [content of which is X]' — 'a promise that X.' までに marks the deadline for being home.