I live in Tower Hamlets, the London borough with the largest proportion of Muslims in the UK. I see plenty of women every day who wear a veil of one kind or another. I don't have any kind of Muslim background so what could I do to start understanding why they wear what they do?
I went on a book hunt and luckily I found a book that gives a really clear background: "A Quiet Revolution" by Leila Ahmed. It's a book that describes some of the twentieth-century back-and-forth of different Islamic traditions, trends and politics, and how they relate to veils. The book has a great mix of historical overview and individual voices.
So, while of course there's lots I still don't understand, this book gives a really great grounding in what's going on with Muslim women, veils, and Western society. It's compulsory reading before launching into any naive feminist critique of Islam and/or veils. I'm sure feminists within Islam still have a lot to work out, and I don't know what the balance of "progress" is like there - please don't mistake me for thinking all is rosy. (There are some obvious headline issues, such as those countries which legally enforce veiling. I think to some Western eyes those headlines can obscure the fact that there are feminist conversations happening within Islam, and good luck to them.)
A couple of things that the book didn't cover, that I'd still like to know more about: