TL;DR Being transgender is normal and medically sound. Transphobia hurts real people, badly. Transgender people in sports is a tough topic but sports organizations have proposed solutions. Please read the last paragraph of my post, if nothing else.
There are a lot of strong claims and language going on in this thread but very little real discussion of the physiological and scientific basis of what being transgender means. Hopefully I can clear up some of the misunderstandings present in this thread.
For starters, basically every relevant medical organization recognizes being transgender as a legitimate thing to be, not as a mental illness but an identity. That includes the World Health Organization, the American Medical Association, the American Psychological Association, the American Association of Pediatrics...the list goes on. If you're someone claiming that "the science" supports that being transgender is made up, you're disagreeing with most current experts -- either you're less informed on the evidence than they are, or you have some valuable research you haven't shown anyone else.
And I meant it when I said being transgender is not a mental illness. Gender dysphoria, roughly the feeling of "being trapped in the wrong body," is kind of considered one (it was reclassified into a middle ground status last year), but its treatment is therapy and hormone treatment; the solution to gender dysphoria is to let the person actualize the identity they feel. So if you're calling transgender people "mentally ill," you are medically wrong.
But what about chromosomes? You can't possibly be a man if you have XX, right? Well, my hypothetical friend, you're confusing "sex" (a biological state) with "gender" (a sociological state). This is a completely accepted distinction in social science where how you express your identity is tied to the molds that society assigns to different sexes (i.e. "gender roles") are not the same as the sexes themselves. Speaking of chromosomes, there are many more exceptions to the binary XX/XY system than you think. There are women with XY who are testosterone-insensitive and may live their whole lives without know they are technically "male" in sex (at least until they take a fertility test). Or intersex people who are born with parts of genitals of both sexes, in which their gender is most likely decided by the identity they "feel." Basically, chromosomes aren't enough to determine anything definitively.
Also, I see a lot of misconceptions about how identifying works in trans people. In an overwhelming majority of cases, they don't simply "decide" to switch and put on a suit or dress on a whim, nor have I ever heard of someone faking being transgender for some sort of competitive or other kind of advantage. These are deep-seated feelings that sometimes be traced back all the way to kindergarten (according to research, gender identity is usually established by about 4 years old and very rarely changes). For many, it involves years of painful questioning of personal identity and often rejection from family, society and the self. To reduce this challenging life experience to "Boys called themselves girls for a day and are ruining a sport" is pretty ignorant and apathetic toward real people.
And all the while that we're discussing the legitimacy of transgender people's existence, they are getting harmed in the real world. Transphobia, as seen on this thread intentional or not (e.g. anyone who called the transwomen racing "men" or "trannies" or implied transwomen are a threat to ciswomen in bathrooms), prevents perfectly functioning transgender people live their lives, as they bombarded by bullyingm rejection and often violence and murder (contrary to the myth that transwomen people are male creeps trying to abuse women, transgender people are in fact much more likely to be sexually assaulted than a cisgender person). This physiological conversation in the context of sports is an important and challenging one, but don't spread more of the hate that causes real damage to real people.
As to transgender people in sports, and it's a tough question without the clearest answer now. The current stance of the International Olympic Committee requires sex reassignment surgery, legal recognition of their gender and two years of hormone therapy. The NCAA only requires one year of testosterone suppression for transwomen. I will end on a quote from the medical experts consulted by the NCAA, which I think is crucial for anyone in this thread to read:
"It is also important to know that any strength and endurance advantages a transgender woman arguably may have as a result of her prior testosterone levels dissipate after about one year of estrogen or testosterone-suppression therapy. According to medical experts on this issue, the assumption that a transgender woman competing on a women’s team would have a competitive advantage outside the range of performance and competitive advantage or disadvantage that already exists among female athletes is not supported by evidence."