Predictably, Maya's answer is clear and accurate, while Robin's is full of straw men and absurdities, like this:
When, for example, citizens use changing or toilet facilities in a shop, we do not provide our birth certificate or take a chromosome test. Rather we are judged on the outward expression of our sex category, our gender. That is why the statutory guidance says that service providers should normally accommodate customers according to their gender
Robin has said before that Robin believes that access to such facilities should be based on things like clothing - see Robin's blog from 2010 when Robin decided whether to use the male or female toilets according to what clothing Robin was wearing at the time.
The rest of us just assume that most people will be decent enough to use the correct facilities for their sex. Similar to the way that we assume that most people don't go into a shop in order to steal. But this just shows Robin's low opinion of transgender people - that many or most of them would try to use the opposite sex facilities if they thought they could get away with it, with no consideration for other users, rather than the correct ones for their sex.