Trans can mean change, as in transform. Some transgender folks choose to change their bodies to become the gender they have always felt to be. The word that describes this way of being transgender is transsexual. A transsexual is a person who has been born into one gender, but identifies psychologically, emotionally and socially as the other, and chooses to live in the world accordingly. Sometimes this includes hormone therapy and surgery to create the physical body that matches that gender.
Transsexuals generally feel that their biological sex is a mistake.
Often they have been aware of feeling trapped in the wrong kind of body since they were very young, and the process of altering their physical self to match their emotional self is tremendously liberating. This may sound obvious, but we need to remember that transsexual people are the gender that they have decided to be. In other words, a person who was born with a female body, who decides to become a man, is then male. A person who was born into a man's body and then decides to become a woman, is then female. But when we are aware that someone is making a specific choice about their gender identity, we should honor that.