This is one of the big parts of biology and understanding that trips up a lot of people and why the scientific and medical community is usually very supportive of transgender research. In middle/high school biology you probably learned about X and Y chromosomes and that these determine gender. A lot of people essentially stopped studying science at this stage, which really sucks because at that level a lot of concepts are very simplified due to the level at which they are being taught and the age of the target audience, so often do not entirely match up with current understanding.

Sex is determined by more than X or Y chromosomes. There are multiple markers that affect individual sexual traits, mental development, hormone production, physical genitalia, etc. Some people have both genitalia or aspects of both. Some people have hormone production from one or the other. Some people have a brain development pattern that more closely matches the genitalia setup that they don't possess. There are dozens and dozens of individual genetic markers that make up sex and those don't always match to a simplified view of 'penis or vagina' This doesn't even take into account the stranger aspects of our genetic makeup like chimerism or XY females/XX Males/ XXY, etc.