Many people take a dislike to the word they or their being used to refer to a single individual (as in “You say you met a rather interesting individual today? Could you tell me more about them?”) or with antecedents that normally take singular agreement (as in “Could everyone raise their hand, please?”).
This usage tends to be referred to as “singular they” and it has a most venerable history (it was used by Shakespeare and the translators of the King James Bible, among many others, and continues to be used by respected writers and speakers). It is also clearly useful: by using “they” in such circumstances, we don’t have to use a word like he or she that implies a particular gender. Some people prefer not to employ it, and are welcome to their preferences. To accuse those who do of “grammatical incorrectness”, however, betrays either a failure to understand what grammar is or an ignorance of English.
Whatever their origin, the relevant assumptions seem to be, first, that any noun or pronoun in English must belong exclusively to one of two categories (singular and plural), and, second, that a pronoun in one category must not be used to refer to an antecedent in the other.
Now, these assumptions look rather a lot like some assumptions that occur elsewhere: specifically, that every human being must belong exclusively to one or two categories (male and female). There’s a further, closely related, assumption that any person with two x chromosomes should be referred to as “she”, while a person with an x and a y chromosome should be referred to as “he” (even if this contradicts their wishes, or their own sense of identity).
Which leads me to think we should expect a very high correlation between transphobia and opposition to singular they. Not only does singular they itself violate the exclusive-category assumption noted above, but it also allows a speaker to avoid assigning human beings to the male or the female category. I think there’s a clever experiment waiting to be devised.
Some readers may point out that this usage leads to “they” being ambiguous, since it will not be clear whether it is being used to refer to one individual or several. This is true, but not a very good argument against employing the usage (and no argument at all for calling it “ungrammatical”): cases of genuine ambiguity are rather rare, and there are fairly straightforward ways of avoiding them when they arise.