Women in Victorian England campaigned for their own bathrooms because they wanted a safe and private place to do their business. Originally there would be a single public bathroom with no gender marking, but what that really meant was that they were men's bathrooms, and women would wait to go at home. This was a pretty big damper on their ability to participate in the public sphere because it limited how much time they could physically spend in public. Gendered bathrooms followed later in other countries for similar reasons. They exist for the same reason that trains in Japan have women-only cars, why there is a women-only rideshare app, etc.

https://www.livescience.com/54692-why-bathrooms-are-gender-s...

http://theconversation.com/how-did-public-bathrooms-get-to-b...

https://en.wikipedia.org/wiki/Sex_differences_in_crime