It's an interesting question to explore, and whichever way I look at it, it always comes back to psychology.
I try and frame it using my own experiences. When I was a teenager I was deeply depressed, and I very much wanted to be male. I remember reading one of Poppy Z Brite's (dark author) articles about why she felt she was a man. I'd not really come across anything like it before, and for a while it stuck in my head. I pondered on it myself: could I stop being a woman and start being a man? What would change? I looked into transsexualism, I read about operations, I listened to interviews.
The conclusion I came to was that I couldn't be a man, because I was not born male. I could take all the hormones in the world, I could fashion a penis from the skin of my thigh, I could grow a beard, and I may even eventually "pass" as male to strangers in the street. However, I would always know the truth, and that truth is that I was born female, and no matter how much at the time I wanted to be male, all I would be doing is lying to myself. I've never been all that great at that.
Fast forward, through much more introspection and some counselling (for unrelated issues) and it became clear to me that I never once "felt" like I was male, because I have absolutely no idea what that is. Because I was born female. What I wanted was to get out of what I perceived to be the female prison I had been born into. I had been sexually assaulted by men, I had received multiple threats of violence from men. I had grown up in a household with a very old-fashioned, very misogynistic father and brother who expected me to be a stereotypical girl, both in appearance and demeanour when I was anything but. I refused to wear skirts or dresses before I hit five, much to my parents' distress. I was outspoken, I was opinionated, I was clever. I wanted to be more than my given lot as a girl.
It took until my 20s to feel comfortable being female. Not because I thought I was male, but because females get such a shit time of it in society and that was echoed and illuminated in my own upbringing and my own social sphere. I even imposed it on myself, or rejected any trappings of femininity because it felt weak. There have been many times since when something has happened, either politically or socially and I've thought "bloody hell, if only I was male then this wouldn't be happening to me" but now, instead of wishing I was someone else, I fight for change.
So from my experiences above, I can see why so many people think they are the opposite sex. It's easy to take a leap and believe in something without evidence (look at religion) when you are dissatisfied with what you have. It's easier to believe you are wrong (we are raised to always blame ourselves) than it is to think that the world around us needs to change. I can see why more and more girls are transitioning too - what girl wants to be used and abused in this porn-sick society they've been brought into?
I've written loads so I'll draw this to a close, but I also see heavy parallels with the fan fiction crowd (stay with me) - women readers and writers who exclusively read and write about gay male relationships. With male/male partners they get to redefine roles and explore stereotypes. It's not Mills & Boon they are reading because they don't want to be the fawning, flouncing female on the cover, and it's not heterosexual erotica because they don't want the female role handed down from porn. It's male/male relationships and love, because they get to explore a shift in power dynamic they would never get to experience in their own lives; they get to "identify" with the symbol of ultimate power we are given in society: the penis.
I don't blame people who experience gender dysphoria for this, I blame society. And until we abolish the notion of gender and do away with these bullshit stereotypes, this will continue. It's getting worse, in our social media age. We need to examine why.