One reason girls need a "girls only" space is because it gives them a place where they can take on roles with gender being less of an issue - for instance, girls are more likely to be called bossy when they take on leadership roles, boys are hardly ever called bossy - the "Ban Bossy" campaign does good work in this space: banbossy.com/. In a "girl only" space, if a leader is required, it's going to be a girl, so the "bossy" label is less of an issue.
This helps level the playing field. If you look at the world the girls are growing up in, that's important. For example, males take lead roles in kids books, tv and in primary schools most headteachers (leaders) are male, while most teachers are female (supporting cast). Not all, obviously - but in general.
We have a female Prime Minister now, but it's been shown women are generally given positions of power when the situation is such that whoever takes it on is expected to fail or come out badly, e.g. a time of crisis or high risk (Google "glass cliff" for further info). That said, I think it is still a good thing to have a role model of sorts in that role, whatever your political views.
I think/hope things are changing, slowly, so at a certain future point there may be no need to have a "girls only" club, or even such that "girls only" and "boys only" clubs aren't an issue. I really don't think I'm a "man hater" but males are less restricted in or judged for their choices than females - this judgement may at times be explicit but is often implicit through the media they are exposed to, which drips the idea that men are more important than women and females are there to perform specific roles e.g.:
Study finds huge gender imbalance in children's literature: New research reveals male characters far outnumber females
Women In The Media: Female TV And Film Characters Still Sidelined And Sexualized, Study Finds
Tween TV reinforces sexist stereotypes study claims
Almost half of 2015 films fail the Bechdel test - the Bechdel test is whether the film has a) at least 2 named female characters who b) talk to each other and c) the conversation is about something other than a man.
So, I also agree with the poster who said if we lived in an equal world, boys should be able to go to brownies, but at present we don't.