Father wears a dirty,oil-soaked monkey suitthat cuts him under the arms, (7-9)
The apparent head of the family, and head of the station, wears a dirty suit that's too small for him. This might mean he doesn't care about what he looks like, or more likely, that he can't afford a better suit. This tells us the family might be a poor one.
and several quick and saucyand greasy sons assist him (8-9)
He's got sons! Several of them, and they're real slick kids and they're helping papa run the family business. We have a feeling that someday, they'll be sporting an ill-fitting monkey suit, just like papa.
(it's a family filling station), (10)
The speaker is perhaps stating the obvious at this point. Maybe that's why she's tacking on the parentheses here.