Gender identity, simply put, is a person's innate sense of their own sex. Now, if you are one of the minority who either don't have, or are not aware of their gender identity, then this can be a difficult thing to understand. You're like a person who was born blind trying to understand what red looks like.
However, don't let that put you off - we can't see electrons either but we know they exist because they are suggested by science. So what science is there to suggest that gender identity exists? Well here's one for you:
There is a birth deformity called "cloacal exstrophy" which involves severe malformations of the lower abdomen. In the past boys born with this condition were often given a "sex change" shortly after birth and were raised as girls with no knowledge of their male past. Despite this, a large percentage go on to express a male gender identity.
That innate knowledge of their own sex is gender identity.
www.ncbi.nlm.nih.gov/pmc/articles/PMC1421517/
One thought experiment you could try is to imagine what it would be like if an ingenious neurologist transplanted your brain into a man's body. Would you now feel like a man, or would you feel like a woman inside a man's body? There's no right or wrong answer and if you are genuinely "agender" (without a gender identity) then the experiment is not going to tell you anything. However, for some people, it's an experiment through which they start to understand.
It's a thought experiment so it's not about current medical science, however, the boys with cloacal exstrophy kind of did have their brains transplanted into apparently female bodies, so it's a good analogy. And as you can see, many of them felt like boys in female bodies.
Most of you will of course, dismiss this. That's OK - doctors and scientists don't doubt the existence of gender identity, and you'll make little progress until you acknowledge it.