Because society establishes that someone born in a certain sex, i.e. man/penis/XY chromosome, woman/vagina/XX chromosome, should dress a certain way, talk a certain way, walk a certain way, like certain things, shave certain areas, not shave certain areas, etc. Biology doesn't establish what clothes a person should wear, society does. Biology doesn't really establish how sensitive a person should be, society does. Biology puts in place how certain clothes look on certain people and how sensitive certain people are, but there is nothing in nature that says that every person of a certain sex should wear only certain clothes or that every person of a certain sex should be at this X amount of sensitivity. Society sets those limits.
Transgendered people feel that they fit better in the gender role that society has said does not fit with their physical sex. It isn't that hard to figure out. That is why most professionals who talk about sex and gender do not use them interchangebly, from what I have observed. Sex is used to describe a person's physical features that make them a man or woman, while gender is generally used to describe the person in relation to whether they fit into society's description of how a man or a woman should act, dress, feel, think, etc.