To be honest, I've been fat, and I've been not-fat, and I feel more like a "real woman" when I'm not fat than I did when I was fat. When I was fat I felt like a very unattractive blob.
I think they only say things like "big is beautiful", or "I'm a real woman" to make themselves feel better about the fact that they actually aren't very comfortable in their own skin. I mean, I could be wrong, but it seems as though they're trying to talk themselves into feeling better about being overweight. Hiding behind a facade of "it's great to be a big woman" isn't a very good substitute for actually losing the weight and being a more normal size though, but that's just my opinion.
I think people have taken it way too far. In my opinion, this sentiment was first meant to mean that a "normal" sized woman was "real", as opposed to the rib showing skin and bones you see on the cat walk. But people have taken it to mean that an unhealthy weight is being a real woman- in my opinion just to make them feel better about themselves. But now you see women who are clearly an unhealthy size with fat rolls down to their knees saying that they're "real". I have a woman on my friends list on Facebook who is seemingly obsessed with being a "real woman" because she doesn't eat "rabbit food" or whatever. Well maybe if she DID eat a little rabbit food (aka not Big Macs and Krispy Kremes) she'd be smaller than the size 28 she is now.
Men do it too. Look at that stupid Jacamo advert with Johnny Vegas "Proper men's clothes in proper men's sizes" and they do sizes up to 6xl. It's just a way for fat people to feel better than skinny people.