Unlike those who are "cis" gender, the term transgender apply to those who have a gender identity or feel an affinity towards one gender over the other -- and it's different than what is asigned for them, to them, at birth.
But if you think this means a male-assigned individual who feels female or a female-assigned person who identifies or expresses as male, exclusively, you'd be wrong.
Well, not wrong, per se, just without the whole picture: see, the term "transgender" has come to incorporate notions of genderfluidity, which could mean pangender, genderqueer and bigender, amongst others. It could also denote the idea of a "third" gender, including those who are cross-dressers, regardless of gender identity.
Though gender is inevitably tied up, in certain ways, with sexual orientation, as far as the term transgender goes, it's solely to identify and speak about the identity of one's gender. It doesn't really have anything to do with sexuality or sexual preference.
So, a trans man is not automatically homosexual, simply because he identifies as female. This means that it can't be assumed that his sexual preference would be exclusively for other men, women, either or even both.
That makes gender expression a highly personal and highly subjective state and it's up to the individual to express it, come out, and proclaim their desire to be recognized as such.
Trans-sexuality, on the other hand, is a term given to those individuals within the transgender population who wish to transition, surgically and permanently, to the gender they identify with. This means a trans man would want to undergo a sex change surgery and process to "look" or "feel" or even "be" "more female", in his own mind, as well as the minds of others.
This is known as a sex reassignment surgery. And its implications are not simply personal or biological. Those who undergo this process are concerned with wanting to experience the legal rights and recognition of one sex along with the gender roles they can now play with a greater sense of "alignment".
Of course, the media is a double-edged sword. On the one hand, we have shows like "RuPaul's Drag Race" and stars like Laverne Cox playing a role in mainstreaming and normalizing this idea of gender-play, gender-fluidity and the need for a "third gender" to assert their own legitimacy.
On the other hand, there are still stereotypes about the nature of trans individuals as being corrupt or sexually promiscuous, hidden or dangerous. This is exacerbated in non-Western country, where gender and sex are usually highly "regulated" by religion.
There is no doubt a greater prevalence of individuals who are in the spotlight, using their position and notoriety to gain attention for the rights of trans-individuals. People that come to mind are the quirky director siblings, the Wachowskis whose recent transitions have been both embraced and criticized.