Transgender actually does not mean "of the opposite gender"...that doesn't really even make sense there.
The meaning of the term "Transgender" is that you are somehow non-normative in your gender identity or expression, i.e. in some way shape or form, your gender identity or expression do not match the gender identity you were assigned at birth by a doctor/your parents. (and yes we are all assigned one of these in our society, its the gender marker on our birth certificates and the gender role we are rasied in). Transgender can include anything from "cross-dressing" to changing your body to more match your true gender identity (commonly known as being transsexual).
Being intersexed is a physiological condition that doesn't have much to do with gender identity except that gender identity cannot be simply assigned at birth (though it often is anyway, removing or hiding or downplaying characteristics that don't fit the simple binary). Therefore its very simple for someone to be intersexed & transgender, all they have to do is express a differet gender identity than the one they were handed on that birth certificate, or even simply act outside their prescribed gender role to some large degree.
Also, there are more than just two binary gender expressions/gender identities. There are many people who are in one way or another not just 'male' or 'female' in their own internal gender identity, and these people tend to use these pronouns, as a way of showing they are not simply one of the two binary genders.
To disclose my personal stake here, I have many transgendered friends, and I myself am transsexual, what's known as "mtf".