I strongly suspect that the only thing his conservative family meant by him "moving to the left" was that he was specifically becoming less of a bigot on trans rights (and likely on LGBT+ issues in general).
And if "not being a bigot" (even as a motive for a violent crime) is seen as being on "the left", that's saying more about how bad "the right" is than anything else.
And if "not being a bigot" (even as a motive for a violent crime) is seen as being on "the left", that's saying more about how bad "the right" is than anything else.