By "hiding your age," you mean not telling your age, right? You're not talking about what makeup you use to hide your wrinkles? That's a whole other question.
I was taught that it was impolite to ask people their ages (so she shouldn't have asked you yours). Actually, I'm not even capable of guessing people's ages. But that's just me.
Aside from the fact that at a certain birthday a woman may think, "Good grief - I'm turning into my mother," nobody likes to be perceived as OLD. There's a thing in our culture about youth. Aging is not highly regarded, unless you are a 150-year-old antique vase, and not always then. Look at the "over the hill" jokes around, even when a person turns 40! When you express an opinion, the response may well be, "Oh, but you're so old you don't relate to real life now." There's an aggressively promoted perception that you're out of the cultural mainstream, and that means you're less valuable.
Advertising does this, too. The older you get, the fewer advertisers are interested in you, except for selling retirement condos, medical insurance, and pills for this and that. Older people are not a target group for good-looking clothing except in some specialized catalogues and web sites.
So sometimes women think, "I want to stay in the main part of society as it exists around me. I don't want to be perceived by others as 'over the hill' and useless." And that's why they don't want to reveal their age.
(And I celebrated the 37th anniversary of my 29th birthday only a few weeks ago.)