Before making any argument related to gender or to transgender people, you have to understand gender itself. Unfortunately, gender isn't at all clearly defined because it's about collective social attitudes (which change over time!). Expectations related to gender vary across societies and time periods, and fall into varying degrees of moral black or white.
As a particularly unfortunate example, the idea that women are inferior to men and should be subservient is one such expectation of gender. Of course, in modern society most of us believe that's a totally outmoded view that ruins lives, and anyone can simply look to the feminism movement for extensive reasoning as to why, in what aspects of modern society that is still the case, and ways that we may continue to improve in that regard.
But not all gender roles are quite so insidious. Why, exactly, do we have the notion that men are typically more dedicated to following sports than are women? Why, exactly, is it considered perfectly normal for a woman to wear a skirt, but not a man? Why, exactly, is it normal in some families for the women to hug everyone whilst men shake hands with other men? Why, exactly, do we default to assuming that the average woman is much better at cooking or sewing than the average man? Why, exactly, do we raise our children according to these standards?
There are a plethora of exceptions to every single one of those rules. Yet, there they are. That's the gender binary, only truly definable through example. The gender binary exists, regardless of whether or not any individual person thinks it should, and the argument that it doesn't exist is fundamentally different from the argument that it shouldn't.
And all of that is gender. So what is it about these roles which links them to a child by virtue of a sometimes-arbitrary sex assignment at birth? Well, not much. Nothing about having one genital configuration or the other actually compels a person to act in certain ways. Unfortunately, most of us are assigned such roles arbitrarily before we are even remotely capable of choosing them of our own volition.
Being trans is fundamentally about understanding that you have the power to choose a different role than the one arbitrarily selected for you. Dysphoria, transitioning, and et cetera are simply pieces of the puzzle. Being trans is about choice. A person can make that choice at any point in life, for whatever reasons they have to do so. To others, it simply shouldn't matter why.
The overwhelming majority people prefer to live in society. As such, we are all subject to its biases. One such bias is placing value on the gender binary; masculine men and feminine women are typically viewed as more desirable and perhaps even more important than are feminine men and masculine women. By virtue of that bias, we are inclined to believe that any arbitrary man we might meet wants to be strong and masculine, and that any arbitrary woman wants to be pretty and feminine.
So what if you don't share those standards? Good for you, mold breaker! You can be a champion of breaking down the gender binary. Go forth and do as you will, and don't answer to anybody!
Well, except that you don't think of your life as a political movement, and you'd really rather not have people staring at you all the time.
But there are other options. So you look into them, pick the ones you like, don't pick the ones you don't, and you live in a way that you choose. Congratulations, you're trans.
When people say things like "a man cannot simply decide to be a woman," it's important to scrutinize what exactly they mean by those words. What exactly do they mean by a "man"? Presumably, someone with male genitalia. But perhaps more - perhaps someone who values masculinity and strength and cannot presume to fathom the minds of women. And what exactly do they mean by "woman"? Presumably, the complement.
But what exactly is there linking one's specific genitalia to their having those particular values? Absolutely nothing. So let's clarify; "Someone with male genitalia cannot simply decide to have feminine values." This statement is a lot clearer, and is explicitly false. A person can have whatever values they want to have. Let's try the other three permutations:
"A person with male genitalia cannot simply decide to have female genitalia."
"A person with masculine values cannot simply decide to have feminine values."
"A person with masculine values cannot simply decide to have female genitalia."
The first is a matter of the limitations of our technology, not of gender itself. The second is at base absurd, though it is necessary to point out that masculinity and femininity are in no way opposites. The third is in the realm of sexual fantasy and suffice to say there is plenty of information to be had regarding whether or not a person who falls into such a category should physically transition (i.e. "no," and most do not).
So if all four potential interpretations of that statement are incorrect, unrelated, or nonsensical, it is probably safe to say that the statement itself is in turn false. There are probably other ways to look at that statement, but I'm sure they would fall into the same categories under scrutiny.
Of course, gender runs ever deeper. As it turns out, gender is more like an axis in ten dimensions than it is a binary. And a person's choice of identity is theirs and theirs alone. Their choice of terminology is theirs, as well, and the choice of specificity is often due to a need to have an identity respected rather than glossed over.
In short, the gender binary is a real thing with real effects on people's lives; people can have whatever personal values that they wish, irrespective of the gender role arbitrarily prescribed to them at birth; in the vast majority of contexts, there is much less conceptual meaningfulness in grouping people by arbitrary physical traits than there is in grouping people by social roles and values; and taken altogether, trans people exist, their concerns are legitimate, and there is very little reason to marginalize them save for ideologies rooted in xenophobia.
And above all, a person's gender is their decision, not yours.
