I have a conjecture about part of the answer.
The world is a complicated place. One way in which we deal with that complication, in law and thought, is by representing a complicated reality with a much simpler model. There are lots of examples:
Some people are more mature than others, in one or another dimension. For many purposes we lump all those differences, along with the continuous range of ages, into two categories—children and adults. Doing it that way makes it a lot easier, in law and in conversation, to deal with issues where maturity matters—at the cost, as with any simplification, of sometimes getting the wrong answer.
If we define gender by genitals, hermaphrodites are both male and female, eunuchs in some sense neither. If we define it by DNA, some apparent males are female, some females male. Some are neither XX nor XY, some both. Nonetheless, we continue to classify people, in the law and inside our heads, as either men or women. Most of the time the simplification fits the reality, occasionally it doesn't.
Someone who does not fit our categories is a problem, not because he is doing anything to us but because his existence makes it harder for us to use our simplified models to make sense of the world. The problem only exists if we are aware of it—XXY genetics existed a century ago, but nobody knew about them. Hermaphrodites existed, and were known to exist, but nobody you knew was a hermaphrodite, or if someone was you didn't know about it, so there was no problem for your day to day attempt to use a simplified map to navigate social space.
The biggest example of this problem, one now more or less over among the people I know, was the breakdown of marriage. It used to be that people could be usefully classified as married or not married, which simplified a good deal of social calculation. As it became increasingly common for couples to openly live together without being married, the classification began to break down. That made it harder to figure out whether you had to invite A to dinner if you invited B, whether you were free to court A, and how to briefly sum up your knowledge of the status of A and B when talking with C.
Transsexuals provide a particularly striking example of the problem. If you knew him as a male and now know her as a female, there is a real problem fitting him/her into your mental picture of the world—a problem that shows up in, among other places, my discomfort with using either gendered adjective. I can see how other people might find similar difficulties in fitting into their heads polygamous families, same sex married couples, children with two mommies, and much else.
I am not, of course, arguing that other people have any obligation to make their lives fit my picture. Maintaining my map of the world is my problem, not theirs—reality has no obligation to conform. But I think the discomfort which comes when reality changes in ways that make obsolete what used to be an adequate set of simplifications provide at least a partial explanation for the strength of the response.