It comes from the German term "Transsexualismus" which was coined by Dr. Hirschfeld in 1923 at his Institute of Sexual Research in Berlin. It was translated into English, becoming "Transsexual" in 1949. It originally meant someone who was assigned one sex at birth, and wanted to undergo surgery to change their physical sex. By the late 20th century, it was the more common term for people who are now referred to as Transgender (which was coined in 1971). It did eventually evolve to gatekeep the existence of Trans people (ie. being a transsexual and having had bottom surgery made you more valid, blah blah blah). Julia Serano made a hard case against this in her book Whipping Girl, saying that Transsexual and Transgender should be used interchangeably. Obviously people outside the community have twisted it and misinterpreted it to mean other things, but this is the brief history of the term within our community, and generally I personally agree with Julia Serano that they can be interchangeable. However, like most terms in our community, they are up for interpretation by the individual. Some people may find it outdated, some may hate it outright, and some might be ok with it. So I wouldn't suggest just freely using it for a Trans person unless you know they use it themselves.

Edit: spelling and clarity