I should preface all of this by "to the best of our knowledge..." or "the evidence very strongly indicates that..." or similar weasel-words, rather than making flat statements. All I can say is that of the competing theories, this one is the only one consistent with all the observations. Where it has holes, there's plausible mechanisms postulated, and competing theories either have to invoke unique mechanisms found nowhere else in Science to cover their gaps, or more often, just plain ignore them.
OK, so what is transsexuality?
Simply, having a gender identity that differs from societal perceptions of the body's gender.
This definition in terms of "societal perceptions" covers the many cases where the body is Intersexed, or where surgical intervention shortly after birth reverses appearance of natural genitalia, or the various Intersex conditions that give rise to sex-reversal syndromes after birth.
For example, a boy who as the result of a botched circumcision ends up with no penis may be surgically altered to an apparently female form (I can think of 2 cases immediately). If the boy revolts against the outside assignation, he is deemed transsexual. The David Reimer case is well known, but Dr Zucker (that's PhD not MD) intervened in another case, and the patient as the result now identifies as a rather butch bisexual woman with severe difficulties in forming relationships and chronic depression. This Dr Zucker deems a "success", as he states on a video available at Youtube.
OK, so what causes transsexuality, this differing gender identity? How do we know that we are male or female?
We're not completely sure, but Milton Diamond's Biased-Transaction theory is definitely proven for some cases. It may be true for everyone. Basically, hard-wired biasses, propensities, instincts, reflexes, and emotional rather than rational responses follow a BiModal distribution. One pattern peaks in females, the other peaks in males, though it's fuzzy. We "know" that we're boys or girls by comparing ourselves with others. "I do not think like X,Y, and Z. X,Y and Z are girls, so I cannot be a girl". Later, this gets extended to "I think like A, B and C. A. B and C are boys, and I know I can't be a girl, therefore I must be a boy". Such mostly or wholly subconscious comparison is usually reinforced by cues such as similar body appearance, social role, clothing and so on, but when these contradict the emotional and instinctive data, they are ignored.
So what causes these hypothetical biases?
This is one area we now have good, and improving, data on. It's all in the lymbic system of the brain, the hypothalamus and nearby structures. These are strongly sexually dimorphic, one common pattern for males, a quite different one for females. You could even say that this is the area that defines ones sex, rather than chromosomes, bodily appearance etc. For transsexual people, the neuroanatomy is strongly towards one extreme, which is discordant with societal perceptions of their "true" gender. Many people of both sexes are potentially "BiGendered", and could function with various degrees of adequacy in either social role, but perhaps 2/3 of people could not.
How can the lymbic nucleus become inconsistent during gestation?
We don't have a good handle on this. We do know that administration of certain hormonal drugs during pregnancy increases likelihood of Transsexuality 500 times. We know that certain gene sequences are associated with increased rates of transsexuality, with different sequences being involved in Male-To-Female as opposed to Female-to-Male transsexuality. The evidence is consistent with the theory that both one of a number of genetic propensities is needed (and we haven't identified more than a few yet), plus anomalous hormonal environment in the womb. Both circumstances are needed, they are both independant, both quite common and normal, but the combination is relatively rare.
The evidence is summarised in BiGender and the Brain, though as the article is over 6 months old, much of the most recent data that confirms the conclusions in it isn't included.
What is Transphobia?
For very good reasons to do with evolutionary development, anything to do with confusion about gender identity tends to cause instinctive and unreasoning disquiet, discomfort, and even revulsion. This can manifest as bigotry and hatred, but most often results in people "not wanting to think about it", and ignoring the evidence saying "I don't buy this whole woman trapped in a male body stuff". Conversely, because the neuroanatomy is bimodal rather than binary, some people have an instinctive partial understanding of the situation. While not transsexual themselves, they have elements of transsexuality in their psychic makeup, and could probably function reasonably well in an opposite gender role.
Transphobia is a perfectly natural instinctive reaction. But so is Xenophobia, Racism and many other evils. Transphobia and to a lesser extent Homophobia are the result of evolutionary pressures towards increasing chances of procreation. Many such beliefs are rationalised afterwards: the instinctive revulsion comes first. Fortunately, we're rational, and not slaves to our instincts - unless we want to be.