It's because there are personal endings depending on the subject. They kind of act like the present form of "to be."
Ben kadınım. = I am a woman
Sen kadınsın. = You are a woman. (singular informal)
O kadın. = She is a woman. (no ending, same for he/it)
Biz kadınız. = We are women.
Siz kadınsınız. = You are women. (singular formal or general plural)
Onlar kadın(lar). = They are women.
and I bet you are a native Turkish speaker :) "I am woman" is wrong in English. "Ben bir kadınım" is unnatural but still can be used in Turkish. Therefore the translation of "Ben kadınım" has to be "I am a woman" and the best translation of "I am a woman" is "Ben kadınım", and the alternative is "Ben bir kadınım"