Why do my boob hurt with my cycles sometimes? What do my hormones do to cause that?

Breast pain. Breast pain is a common problem with menstrual cycles. It can be due to cyst or from hormonal changes some women will have it earlier on their cycle due to the estrogen levels but later on in the cycle is due to less progesterone. You can change your birth control to help, try oil of primrose or vitex to help balance the hormones. Also decreasing caffeine, vitamin E also helps.
Common problem. Breast pain that varies during a cycle is pretty common, and is usually not worrisome. Reasons vary but common ones include breasts swelling and becoming heavier/tender due to water retention; women with fibrocystic breasts often have cyclic pain from swelling and increased number of small breast cysts.