"The style to which she has become accustomed" means "her standard of living."
It's old-fashioned language that has become a joke.
It occurs in Victorian-era novels. It might be used by a woman's father, talking to a man who wants to marry his daughter. It means "are you as wealthy as we are? Will you be able to maintain her standard of living?" It would be part of a number of question that amount to "are you our social equals?"
It suggests that someone is paying close attention the financial aspects of the marriage.