I don't think there's actually one particular reason why I believe in a woman's right to choose, so I'll list all of them:
1) People should be allowed to control their own bodies, and their bodies should not be subject to the beliefs of whoever is in power at the moment.
2) I don't think teenagers should have kids unless they really want to. People are flawed. Not everyone is going to use birth control every time they have sex. And teenagers are particularly likely to make that mistake. One can argue that teenagers should be told not to sex in the first place, but clearly many of them are still going to. Just because a 15-year-old makes a bad decision--or accidentally uses a condom incorrectly--doesn't mean she should be forced to have a baby. I know that if I had gotten pregnant as a teenager, I would not have been able to handle a pregnancy or the responsibilities of motherhood. That's despite the fact that I had (and have) incredibly supportive parents with the ability to help me financially. I can't imagine what it must be like to be a pregnant teenage girl in an abusive or cash-strapped household.
3) Illegal abortions are dangerous and can be deadly. Even if abortion was illegal, women would still get them--especially desperate women who see no other option. And desperation equals vulnerability. By keeping abortion legal, we keep women who need abortions safe. They can have the procedure done by licensed doctors at regular medical facilities, and they can get any and all follow-up treatment they need.
4) No woman should have to carry a child she doesn't want, and no child should be unwanted at birth. I think "every child a wanted child" is an incredibly powerful slogan, and I fully endorse that idea.
Those are my reasons for being pro-choice. I've never had an abortion myself, but I know women who have, and, honestly, they all believe they made the right decision. I'm just happy that it was their decision to make.