I'm a native speaker and I had no idea what that meant either. I just thought 'but carpets and curtains aren't hair -odd' and ignored the bit in brackets. I also had no idea what 'cis-privilege' was and am surprised that asking about it here got you such an unhelpful response. It's quite tricky to google for too. I got two pages of texts _using_ it, but none of them apparently explained what it - meant - clearly it was/is a common term in that sphere of discussion (trans-gender issues).
Eventually I found the wikipedia page 'cisgender' which explains that term, from which one can kind-of guess what 'cis-privilege' is about. Given this non-obviousness, it would have been nice if someone had just explained things rather than saying 'don't be rude - go read the internet'