Is it that murky? I saved the following from a comment on the same issue in a different thread:

Everyone has a gender identity, but most of the time it matches their external appearance perfectly. When it doesn't, that person is described as "trans".

Trans people don't change their genders. Their gender identity already exists, that's something you're born with. Trans people just change their lives to better match it. This often includes both social and physical changes. Trans people also have all the variation in sexual orientation that cisgender (ie, non-trans) people do. A trans woman may be lesbian, a trans man may be gay.

For how gender identity forms, and why some people's gender identity doesn't match their external appearance, that's not entirely understood. But it does appear to be strongly influenced by hormone levels during gestation, which shape the neurological structures of one's brain.

Basically, all fetuses start out physically the same - and that base form is essentially female. Most of the time, if a Y chromosome is present it will be activated by testosterone in the mother's blood. It will then cause the fetal gonads to develop into testes, which then flood the system with testosterone that cause the cascade of changes turning a female fetus into a male infant. And most of the time, if no y chromosome is present these changes won't take place and the fetus will develop into a female infant.

But there are a lot of things that can cause development to go off-script. Hormone variations are one of them. If hormone levels are atypical early in gestation, that can affect external development and the infant may be born visibly intersex. If hormone levels are atypical later in gestation, when the body is basically done but the central nervous system is forming, it can affect neurological development.

The brain is wired to recognize one's body. That's why an infant can pull their arm away from painful stimuli long before they consciously know what an "arm" is. They don't have to learn that their arm is theirs, that knowledge came hard-wired.

Sexually specific aspects of one's anatomy are part of this neurological map too. Vastly oversimplified, a brain that grows under hormonal conditions more typical to a female fetus is going to be wired to expect and control a female body, and a brain grown under conditions typical to a male fetus will be wired for a male body, even if the body it's actually in doesn't match.

When that happens, it is one hell of a mindfuck. This is not a mental illness; the brain in question is working perfectly normally, it's just being subjected to extraordinarily disturbing circumstances. Transition fixes those circumstances and alleviates the mindfuck.

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Citations on the neurological science of gender identity:

An overview from New Scientist

An overview from MedScape

Prenatal testosterone and gender-related behaviour - Melissa Hines, Department of Psychology, City University, Northampton Square, London

Prenatal and postnatal hormone effects on the human brain and cognition - Bonnie Auyeung, Michael V. Lombardo, & Simon Baron-Cohen, Dept. of Psychiatry, University of Cambridge

Sexual differentiation of the human brain: relevance for gender identity, transsexualism and sexual orientation - D. F. Swaab, Netherlands Institute for Brain Research, Amsterdam

A spreadsheet with links to many articles about gender identity and the brain.

Here are more

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Citations on the incredible benefit transition has to the psychological and emotional health and functionality of trans people:

See /u/Chel_of_the_sea's post here

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Citations on transition as the only effective and appropriate medical response to gender dysphoria, as recognized by every major US and world medical and psychiatric authority:

Here is the American Psychiatric Association's policy statement regarding the necessity and efficacy of transition as the appropriate treatment for gender dysphoria. More information from the APA here.

Here is a resolution from the American Medical Association on the efficacy and necessity of transition as appropriate treatment for gender dysphoria, and call for an end to insurance companies categorically excluding transition-related care from coverage.

Here is a similar resolution from the American Academy of Family Physicians.

Here is one from the National Association of Social Workers.

Here are the treatment guidelines from the Royal College of Psychiatrists, and here are guidelines from the NHS. More from the NHS here.