Apparently, Jenni Murray didn't write the title, which has the interesting phrase 'don't call yourself a real woman', and also the idea that having lived as a male disqualifies you from being a woman. I don't really know what 'real woman' means, nor who is going to decide this.
The comments below follow well-trodden paths - there are those who recommend a biological approach, in other words, a man has a penis, and a woman has a vagina. This seems to ignore the social nature of identity - I don't inspect people's genitals, actually. In other words, 'men' and 'women' do a kind of self-presentation.
I suppose some women are angry that trans women want to be treated as women. It doesn't bother me, really, nor the idea that trans men want to live as men.
But it is all very interesting, as it raises many issues about sex/gender, and identity itself. There seems to be a kind of historical sequence, women fought for equality, gays and lesbians had to throw off centuries of prejudice, now trans people are doing the same. I wonder if there is anything more to come?