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Actually, science does a really poor job of answering questions about gender. On the surface, things look really simple: you're either XX (female/woman) or XY (male/man). Genetically, there's actually a lot more variation than that. For example, a non-trivial proportion of the population is born with different sex chromosomes (XO or XYY, etc), ambiguous genitalia, or even bodies that don't match their genes (XY females, for instance). So, there's all of this genetic complexity, but gender is so important, culturally, that we insist on placing people into two distinct categories.
But even aside from genetic variation--even if all people were born XX or XY and had the expected matching body--many have argued that our society creates many of the gender differences that we attribute to biology. As a mundane example, blue used to be considered a sissy color, appropriate only for girls. The fact that this has changed shows that ideas about gender have changed. Also, women human beings adorn themselves and wear makeup to attract mates, which is the exact opposite of what happens with males in the wild, which are often more pretty or fancy to attract female partners. Yet beauty and femininity are very closely linked our conceptions of gender. In other words, the gender differences we consider innate/biological, may actually be things that human being created through culture.
To follow this train of thought, if any parts of gender identity and gender presentation are rooted in culture, rather than in unchangeable biology, then it makes sense that being born XX or XY is not sufficient to make people want to live as women or men, respectively. Though cultural ideas have a lot of force, and probably even more so if they are layered on visible biological differences in bodies, they're not going to resonate with everybody. So, we see kids and adults who feel like they're born into the wrong body.
Are they sick? I don't think so. Why does somebody have to be born XX to identify with femininity?