IMHO, there are only 2 human sexes, along with a small number of abnormalities. As far as I know, evolution of vertebrates has never thrown up more than 2, although there are some creatures that can change their sex e.g. Clown Fish. 2 sexes is fundamental to the way animals propagate themselves. This may of course change in the future, but for now, this where we are.
Gender is a human construct based on perceived differences in roles between the 2 sexes. This perception varies according to culture and time, and the fact that that there are physical differences between the sexes e.g. woman bear children and men don't. In an ideal human world, there would be no difference between the gender roles, but that isn't likely to happen any time soon, however much we would like it to.
The consequence is that right now humans cannot change sex. Not completely. But we can change, or attempt to change, how we are perceived in society with regard to our gender role. How effective this is depends on which culture we live in and how strongly the gender roles are perceived by that society.
I cannot comment on why people feel they are a different gender to their biological sex. I don't understand it and have no direct experience of feeling this way. I don't even know what someone in this position believes the 'other gender' to be, as it varies according to their society. But while I do believe that you cannot change sex, since gender is not entirely based on your physical sex, I don't see why you can't attempt to change your perceived gender.