If Mary was conceived without sin so that Jesus would be without sin, wouldn't that mean Mary's mother would have been conceived without sin as well?
No, because the Immaculate Conception was not necessary for the Incarnation to take place, but it was fitting. The all-powerful God could have become a man in any number of ways, including being born of a sinful woman, since through his omnipotence he could still keep himself free from sin.
However, it was fitting that the Mother of God would be preserved from sin just like her Son. In a like manner, a king could walk among his subjects like them but, given his royal stature, it is fitting that he has a royal procession.
So to summarize, Mary's immaculate conception was a fitting, gracious gift, and the all-powerful God protected her from sin even though she was conceived in the womb of her mother, Anne, who was a sinner like all other human beings.