I recently met a respected Chassidishe person and asked him the question. I hope that I can articulate his response correctly.
He had 2 reasons why a Chassidishe women shouldn't drive.
1) Like many have stated above it starts with the concept of כל כבודה בת מלך פנימה, a Jewish Women is all about modesty therefore the place for a Jewish women is her home. That said society has changed and things are very different in 2018 . The concept however stayed exactly the same. Therefore part of a women's modesty is not to be "out there". While she can come and go freely her primey place is still her home. A women with a car is no longer "a women at home".
2) There are certain jobs/professions that are considered men's jobs. For example Firefighters, Policeman, Construction workers, are considered jobs that are not meant for a (Jewish) Women. It has nothing to do with strength, just the type of job.( Perhaps the word would be masculine type of jobs ( not finness ). While I'm not comparing driving to those jobs the concept is that it's not seen as a woman's place to be.
This was his response. I hope I this shines some light.