You probably need a doctorate in psychiatry or psychology to understand what Gender Dysphoria is but I'm not going to let that stop me. I am simply amazed by the misinformation going around on both sides of this argument.
You cannot change your sex. Sex is a physical attribute that is unchangeable. You can't go into every cell and flip an X chromosome into a Y chromosome nor is there any surgery that I'm aware of that can give someone with XX chromosomes the ability to produce sperm or someone with XY chromosomes the ability to produce ovum. Then again there are some males that can't produce sperm and some females that can't produce ovum so it's not that simple.
But gender is not the same thing as sex. Gender defines societies expectations about the attitudes and actions associated with a given sex. When you have the physical characteristics of a male but the psychology of a female (sorry feminists but men and women do have different psychologies) or vice versa that causes all sorts of stress and psychological issues.
Some people decide to go with their psychology and have reconstructive surgery to change their physical attributes to line up more with their psychology. Others simply accept gender non-conformity as their normal. Honestly, as long as someone isn't negatively impacting someone else, I don't care what they decide to do.
Note that "as long as someone isn't negatively impacting someone else." I think allowing athletes that have gone through puberty as males to compete against athletes that went through puberty as females is negatively impacting female athletes.
The same goes for people who still have male plumbing invading female privacy areas such as locker rooms and rest rooms.