The issue of Transgenderism is intentionally convoluted, so I'll try to boil it down simply.
Transgender theory redefines the words "gender", "male", and "female". Well, to be accurate, the theory strips the words of any actual meaning entirely, and instead reduces the words to a feeling, an opinion.
The theory as a whole has massive problems:
1) It is internally contradictory. On the one hand the theory asserts that there are no concrete, innate differences between men and women beyond the physical. But the theory also asserts that the physical body must be aligned to match the individual's innate, concrete gender. But if nothing is different between men and women beyond the physical, what is the physical realignment aligning to?
2) There is no objective basis to make an assessment. There is no way to assess if someone is transgender without directly asking them. It is less possible of assessment than someone's political identity, as even something as subjective as political identity can be assessed by inquiry into specific policy positions and opinions without directly asking people what their political identity is.
3) The claim is unjustified. It is not possible for a man to know what it is like to be a woman, nor vice-versa. Even the "gender-reassignment surgery" doesn't actually change gender, but instead just makes some cosmetic alterations, amputations and hormones. It doesn't change body structure, brain formation, or relevant organs. It is appearance without substance. No one can claim that they know what it's like to be both a man and a woman. Ergo, it's not possible for someone to know if they even have a preference, let alone know they're mismatched.
4) Many, if not most, activists hold that the cosmetic alterations are sufficient to resolve the problem. Except the core defining elements are missing. Men modified to look like women cannot bear children, do not have periods, are as vulnerable to genetic diseases as unaltered men, produce the same hormones, etc. But the approximation is sufficient for the activists and the sufferers alike. That alone suggests the problem is not a break between mental and physical gender, as appearance alone would not be enough to solve the problem.
5) It demands people suspend their own, factually-grounded, rational judgments in favor of an amorphous standard that basically boils down to an assertion of opinion and a blind wish. Refusal to put someone else's assertion over the hard facts is called bigotry. But accepting a claim as true while knowing it is false is the actual, literal, explicit definition of Orwell's Doublethink.
6) Body Dysphoria is a real thing. It's the belief that some part of the sufferer's body is wrong in some way, and they take great effort and go to great lengths to conceal or correct the flaw. But the flaw does not actually exist. It's a ghost, a phantom.
Body Integrity Identity Disorder is a belief that some part of a sufferer's body needs to be removed, that they shouldn't have two legs or arms, or be able to see or hear. It even includes the desire to be paralyzed. Like the transgendered, people with this illness seek to have the parts amputated, to be blinded, deafened or paralyzed, to have the parts that trouble them removed.
Both of these are clearly psychiatric disorders, and actually neatly parallel transgenderism. Actually, Gender Dysphoria is the clinical diagnosis for transgenderism, and as a result there is a political pressure campaign to have it eliminated as a diagnosis, despite (and perhaps because) of the clear parallels with related disorders.
7) It tells people who are suffering from mental illness, people who are easily influenced or pressured, or even people going through normal identity crises that there is something wrong with them, and that they must correct that issue through extensive, destructive, irreversible procedures.
The simpler and less harmful (if harmful at all) option it to counsel and teach affected people that there is nothing wrong with them or their feelings. They can't know that they are wrong gender because they have no idea what the "right" gender feels like, so such a feeling is irrational and baseless.
The best option, to rephrase, is for people to simply accept who and what they are, feelings and body and all. Being weird or eccentric or different is not bad and does not require surgical correction.
The worst and most harmful option is to indulge their misconceptions and demand that everyone else indulge them too. Avoiding reality is never a good thing.