How does one actually know what it feels like to be male or female? Being male, I just look at myself and say "this is me" and I identify socially as male because of our genetic classification of "male" as well as sharing all traditional "male" interests and activities.

Medically/Psychologically this is called "gender identity".

I've mostly only come across it through studies and I am not an expert on this, but the current feeling is that humans pick up a gender identity at a very young age. Currently thinking is by around 3 a kid identifies with one gender. Now most of the time, a male kid identifies as a male, and a female kid identifies as a female. That can be called gender conformity.

But in that instance where a male child identifies as a female, or vice versa, this is called gender dysphoria. When it conflicts with their social and occupational lifestyles to an enough degree, it's called gender identity disorder.

How I was taught was once this gender identity occurs around that age, it's pretty much fixed. The medical health community has no way to get one to switch the identity afterwards. Just like you look to yourself and say "I am male" you've been doing that since your gender identity occurred around 3 years old and since then it's felt perfectly normal.

The research on what exactly triggers the gender identity and which gender is picked is beyond my level of knowledge, but there is some discussion going on that rather than socialisation (where one becomes a male because one is raised with manly objects and toys) but that genetics as well the hormones that a fetus produces during fetal development may play the role in how the brain is wired to see oneself.