They, like other acidic drinks, strip the enamel off of your teeth. The enamel protects your teeth, so if it strips, it exposes the nerves, which can cause pain when you eat or drink certain foods. I don't know if enamel can grow back or not, but you can always ask your dentist next time you go or do a quick Google search.

I suggest you cut back on the soda. Soda is fine to have once in a while, but not everyday, especially more than once. Soda is full of sugar, chemicals and caffeine. All of this is bad for you, and you really don't want to damage your body. Start off small, having a soda every other day, then cut down to once a week and finally to once a month or so. It's hard to cut back right away, so don't quit cold turkey. Try substituting soda with something less sugary-low sugar juice, milk or water. Try water, since it cleanses your system and keeps your organs alive while also keeping you hydrated. It shouldn't rust or damage your braces, which soda might be able to do. (I don't think you and the dentist will be very happy if you erode your braces due to your soda intake.)