One of the biggest reasons that the word transsexual fell out of favor was the pornography industry. The word became associated, in public discussion, with hermaphodites, futanari, or even women wearing strap-ons. With the birth of the internet, drunk men searching for “transsexual” websites were not the people one wanted trying to throw themselves into discussions on hormones, binders, friendly doctors, etc. Since the term transgender was available and lesser known, and because being transsexual mostly implies being transgender as well, that was the direction that conversation went.
Mind you, this is only my observations; and what I was told when I first started my journey.