OK, I'll give it a try. I'm an economist, so this might be a bit dry.
Any explanation for why women are not entering STEM fields should as well account for the fact that since the 1960's women have entered and continue to enter other highly paid professions such as Law and Medicine. Back in the '60's the proportion of women studying in all of these fields was 10% or less. Now it is 50% or better in Law and Medicine, but lower in Engineering.
So the "chilly climate" explanations would require that senior engineers and scientists in the 1970's and 80's were somehow more discriminatory and more unwelcoming to women than were senior partners and senior doctors. This seems unlikely, as contemporary accounts suggest the first women entering these fields were distinctly unwelcome. The situation for women entering medicine back then was much worse than it is in STEM fields today.
So this leads us to explanations that are based on choice. Why did women choose to storm, and overwhelm, the barriers in Medicine and Law, but not the ones in Engineering? And, now that those barriers are largely (perhaps not completely) gone in Engineering, why are women still choosing to work elsewhere even after twenty years of explicit encouragement?
So here's where the Economics comes in. We need to understand the factors that influence occupational choice. In particular, there is a widespread belief that if two people are equally good at math, then to the extent that abilities affect occupational choice they should be equally likely to enter a math related field. This belief is false.
The basic principle underlying occupational choice was discovered by a guy named David Ricardo back in the early 1800's. He was interested in the patterns of international trade, and the factors that determine what countries choose to produce and export. So, for example, Canada produces a lot of softwood lumber and exports it to the United States. And, if you fly over northern Canada you will see nothing but softwood trees for miles and miles. So Canada must be particularly good at growing trees, right?
No. Relative to the US, Canada is terrible at growing trees. Softwoods grow much faster and better in the Carolinas than anywhere in Eastern Canada. In some parts of northern Canada the growing season is so short it's known as "two months of bad snowmobiling." So why would Canada be sending softwood lumber to a country that can produce it better for itself?
The answer Ricardo provided was that countries do not specialize in what they are good at. They specialize in what they are relatively good at. He called it "Comparative Advantage". The reason Canada grows trees is that there is absolutely nothing else that can be done with that land. In the Carolinas you can build golf courses, orchards, and stately homes made from cheap imported softwood lumber.
So, to get back to women in STEM, suppose you are in high school deciding what to take in university. We know that girls get higher grades than boys in every subject except math, and sometimes even there as well. So think of a boy getting 90% in Math and 70% in English. Or, think of a girl getting 90% in Math and 90% in English. These two will do equally well in a STEM field, but which one is more likely to choose to enter a STEM field?
The question isn't: "Why are there so few girls in Engineering?". The question is: "What else are boys going to do?".
A couple of final points. First, this explanation has been empirically tested. Basically, if you run a regression to predict the choice of college major based on high school grades you will find that a higher grade in Math will make students more likely to choose a STEM field. But if you add Math grades relative to English grades, or math grades relative to overall average grade, the regression performs much better. Nonetheless there is still something left over, in that holding grades AND relative grades constant, men are still somewhat more likely to choose a STEM major. So the comparative advantage story helps, but it's not the whole story.
Second, while women are under-represented in STEM fields, it is men who are under-represented in universities as a whole. If a school does not have an engineering program it's like Surf City. (That's a reference to a Beach Boys tune with the chorus: "Two girls for ev ... ry boy.") So, instead of spending money trying to encourage girls to do math, how about focusing on getting boys to read? This would at the same time lead to relatively more boys in university and relatively more women in STEM. Seem like a no-brainer.