Certain comments are, "Trans women are still male since they will always have XY chromosomes.", "Doesn't saying, '..gender identity is a part of the brain..' fly in the face of the claim gender roles are societal?", "Shouldn't trans people be discriminated against if it's called Gender Identity Disorder?", and "If I'm a tree on the inside does that mean I should be called a tree from now on?" lol some of these people are just being jerks, but some people are genuinely wondering which is why I'll be happy to address these.
In psychology the part of gender identity that makes you identify, and relate with a certain sex is considered to be different from gender roles, even though relating with a certain sex eventually makes many people adopt certain gender roles because society shows this sex assigned to these certain roles. Psychological studies observed that once people reach the age of 3-4 the sex they relate with has solidified. It can't be changed after that, so they know it's not something people choose.  If this was a choice transsexual people getting paid minimum wage would not be spending 30 thousand dollars for surgery. It's something they can't help, and they shouldn't have to pay this much for something they need.
This also suggests that the part of gender identity that makes you relate with a certain sex does have biological causes. Many studies have been done where they found differences in male, and female brains, and they also looked at transsexual brains. What they found is the transsexuals had the same differences of the sex they identify with. This happened consistently across many studies independent of each other. 
Some people see "Gender Identity Disorder" (Which was officially changed to "Gender Dysphoria" recently.), and instantly think, since it's a disorder it means people should stop being transgender, but psychologists are saying the opposite. If transsexuals are made to feel out of touch with their gender identity, by society, or some other means they have shown to go through severe depression, anxiety, or even sometimes commit suicide. So psychologists find the best thing for transsexuals is to live as our gender identity, and to get a sex change.  This is what they mean by gender dysphoria. Psychologists don't simply say a sex change is recommended, they say it's medically necessary for the mental health of transsexual people.  Some transsexual people don't have gender dysphoria because they were accepted by society, and received access to sex reassignment surgery. It's not a meant to describe transsexual people as a whole, just transsexual people who experience extreme dissonance with their gender identity. 
In psychology the difference between gender, and sex is recognized. Gender is what you identify as, and sex is your anatomy. This is what transsexual women mean when they say they are women, and would like to go by she. It does not mean they claim they have ovaries, XX chromosomes, etc. or anything else that is not based in fact. Transsexual women would not try to transition in the first place if they thought they already have a vagina.
These ancient definitions of saying people are simply either male, or female come from a time when we didn't even have the scientific method as we know it, so it's not surprising they don't accurately describe sex in humans. It would be more scientifically accurate if society moved towards using "male", "female", "intersexual", to describe sex, and "he", "she", "man", and "woman" to describe gender. In this way it does not ignore scientific facts. Transsexual women are born intersexual in sex, and women in gender, and vise versa with transsexual men. Describing them as male goes against fact because it ignores the part of their brain that is female, and the secondary sex characteristics that are made when they have a sex change. Transsexuals are able to transition the parts of their bodies that have any significant effect on their daily lives. That's what matters the most for them to live as their gender. When you meet someone regular people don't immediately scan their chromosomes. So chromosomes are not important when considering how you live your daily life, which means it's not going to hurt anything if trans women go by women, or trans men go by men.
Many cultures throughout history have recognized more than 2 sexes in different ways because of the variancy in the sexes. There have been 3-5 sexes categorized by some societies, and sometimes they don't attach a number to it.  The term third gender has been used to describe hijras of India, Bangladesh and Pakistan who have gained legal identity, Fa'afafine of Polynesia, and Sworn virgins of the Balkans, among others, and is also used by many of such groups and individuals to describe themselves.  Many native-american tribes described them as the two-spirit people who were treated with great esteem as shamans. 
A fun fact is it is literally true that every person on earth is either female, or they were for the first two months after conception. No, I didn't make a typo. The physical structure of every human fetus starts out as female. If anyone studies a scientific book on the fetus, the "sex" genes are XY, rather than XX, hormones will be secreted that will initiate a sex change at two months after conception. This means every guy has already had a sex change.
1.) "Gender identity appears to develop in the early years of life and generally is established by age 3 years." - The American Psychiatric Publishing Textbook of Psychiatry, Sixth Edition 2014, page 682.
2.) "Both 3- and 4-year old children reacted in a gender stereotypic manner to conduct by peers that did not conform to their gender. They disapproved of boys feeding, diapering and comforting dolls and girls driving dump trucks. They also expected the peer's friends to react in the same disapproving way.However, the 3-year olds did not exhibit differential self-evaluative reactions to engaging in masculine-typed and feminine-typed activities (Figure 5). Nor did their self-reactions predict their gender-linked conduct. By contrast, the 4-year olds exhibited substantial self-regulatory guidance based on personal standards. They expressed anticipatory self-approval for conduct linked to their gender but self-criticism for conduct deemed appropriate to the other gender.Moreover, their anticipatory self-sanctions predicted their actual gender-linked conduct."
Bussey & Bandura 1999, Social Cognitive Theory Of Gender Development And Differentiation
3.) AA Ehrhardt & JN Wasserheit 1991, Age, gender, and sexual risk behaviors for sexually transmitted diseases in the United States
4.) Diamant, Louis, and Richard D. McAnulty, eds. The Psychology of Sexual Orientation, Behavior, and Identity: A Handbook. Westport, CT: Greenwood Press, 1995.
5.) Golombok, Susan, and Robyn Fivush. Gender Development. Cambridge: Cambridge University Press, 1994.
6.) American Academy of Pediatrics, 1999
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14.) Meyer W, et al, Standards of Care for Gender Identity Disorders-Sixth Version, International Journal of Transgenderism 2001
15.) The Harry Benjamin Standards of Care: http://www.tc.umn.edu/~colem001/hbigda/hstndrd.htm
16.) The World Association for Transgender Health, "These medical procedures and treatment protocols are not experimental: decades of both clinical experience and medical research show they are essential to achieving well-being for the transsexual patient." - http://www.wpath.org/site_page.cfm?pk_association_webpage_menu=1352&pk_association_webpage=3947 - The World Association for Transgender Health is an international association of psychologists, and is backed by the American Association of Psychologists: http://www.apa.org/topics/lgbt/transgender.aspx - They are the founders of the Benjamin Standards of Care, which is the most widespread standards of care used by professionals working with transsexual, transgender, or gender variant people.
18.) Roscoe, Will (2000). Changing Ones: Third and Fourth Genders in Native North America. Palgrave Macmillan.
19.) Graham, Sharyn (2001), Sulawesi's fifth gender, Inside Indonesia.
20.) Martin, M. Kay and Voorhies, Barbara (1975). Supernumerary Sexes, chapter 4 of Female of the Species
22.) Agrawal, A. (1997). "Gendered Bodies: The Case of the 'Third Gender' in India"
23.) Young, Antonia (2000). Women Who Become Men: Albanian Sworn Virgins.