This is a tough question to answer and impossible, I think, to really do justice to. Lord knows people have tried and Ive largely given up attempting to explain it. Its like trying to explain to people what its like to be a purple gazelle. Full of shitty euphemisms and metaphors. Ugh.
But...
What we are talking about here is a medical condition. How and why it comes to be is unknown. Maybe its hormonal, maybe its genetic, maybe its just idiopathic? What we do know is that when you cut up the brain of a trans person, their brain structures match more closely those of their identified gender, not the gender they were assigned at birth.
We also know that there is no way of diagnosing or identifying these changes medically, so we rely on the person to self-identify as trans.
Because diagnosis can only occur in talk-therapy, people assume this is a mental condition. It is not. It has mental/emotional components (depression, anxiety, etc) but you CANNOT treat it as a mental condition. Treating the symptoms does not resolve the dysphoria. The only known way to relieve the actual dysphoria is via transition.
What is dysphoria? Linguistically, it is the opposite of "Euphoria". In practice, it is the full range of discomforts a person can feel around their gender and the various ways gender is understood or expressed.
Because we are talking about a medical condition, we must also talk about the fact that no medical condition presents exactly the same in any two people. There is simply too much variation in humanity to have a universal experience of anything.
Some transgender people feel dysphoria around social aspects of life - how they are treated, addressed, etc. Some people feel dysphoria around physical aspects - their primary and secondary sexual characteristics, feelings of arousal, menstruation, etc. Some people are actually fine with their social situation and even physical situation, but simply do not function well on the hormones their bodies naturally produce - this is often called hormonal dysphoria.
Some people feel dysphoria to only a minor degree while some people feel it very acutely with persistent and debilitating episodes.
How treatment moves forward depends on the person themselves, their level of discomfort and the ways in which it manifests.
Some people can effectively "self-medicate" through short term cross-gender expression. I, for example, was able to reduce much of my dysphoria for many years by giving myself pedicures. Seems silly but it helped for a long long time.
Some people find relief by socially transitioning to their identified gender - clothes, mannerisms, legal status, name, etc. while others seek out medical interventions like hormone replacement therapy or surgeries. Surgeries can be as minor as shaving down the adams apple to create a smoother contour, to removal of existing reproductive organs and reconstruction of genitalia.
It is easy to say "these changes are just cosmetic", but that is not it at all.
Social transition changes the way you move in society. It is not about getting doors held open for you or peeing in the right bathroom, per se. Its the entire sociological engagement with the world. Your very demographic in society. That is heavy shit.
Hormonal transition has external effects, but its more like putting the right fuel in your engine (if you can forgive a shitty metaphor). Sports cars need different fuel than school buses. Put the wrong fuel in your engine and at best, youre going to get suboptimal performance. Change the fuel to the one your engine wants and everything seems to get better overnight.
Surgical transition is the one that tends to throw people the most. Cuz thats the one that involves cutting. "You want to do WHAT to your WHERE?" Its hard to relate to. But your brain is designed to know where your parts are, what they do and how they work together. Thats a massive part of the brains function. To make your parts work together as designed. When your brain has formed in such a way that it expects X body part to be there (or not be there) its presence is upsetting. To use the shitty car metaphor again, youve got a Ford Pinto in the garage and a Mazda Miyata owners manual. Sure, you can say "fuck it, its a car, how different can they be?" but the key goes in the wrong place, the engine is designed totally differently, the tires are different, the car in your garage has a habit of exploding... its just not a good situation all around.
Now, from an outsiders perspective, you can say "sure, the Pinto is a death trap, but its way easier to just get a new owners manual and maybe slap a coat of paint on the rust bucket and be happy you got a car at all" but (and this is why its a shitty metaphor) we live in a world where you get one owners manual; you cant get rid of it, you cant trade it in, and best you can do is write notes in the margins. BUT you CAN replace the parts on the car. All of them? No. Not yet. Youre still stuck with some of the major parts, but you can fix all the rest. You can make that pinto into the miyata of your dreams if you work at it long enough and swap some parts. You can make it so that, with time, that owners manual actually works with the car in front of you. There will always be quirks, sure. The tire pressure might be different, or the seats might be a little further back, but for all intents and purposes, everyone who sees you drive by says "Nice Miyata!". These changes arent just cosmetic. Youre not just changing how it looks. Youre not just changing how it works. Your changing it so your freaking owners manual matches the damn car!
And now Im over 5000+ characters into a post at nearly 3am just trying to make a little sense out of something that is both ridiculously simple and deliriously complex and probably not getting anywhere because IM IN A FUCKING PINTO... and all I want is for there to be a Miyata in the garage.
And if any of this made sense, awesome.