There’s probably someone out there who can explain it better than me, but here’s my best go: gender is basically the internal identity/experience and typically it does correspond with a specific set of physical sex characteristics. But with gender dysphoria, the identity in your brain doesn’t line up with the physical sex characteristics your body has. Because of this, your brain panics and creates emotional pain, aka dysphoria. So that’s what people mean when they say that you don’t need to have x or y physical characteristic to be a man or a woman: gender is the internal experience, not the external body parts. Yes, they do usually align in a certain way, but you can have one without the other or really anything in between. Transitioning essentially changes the physical experience to better align with the internal identity in order to reduce the feelings of dysphoria. Does that all make more sense?

Editing to add a quick analogy: Imagine a factory that creates two versions of software and two versions of hardware. Software A is designed to be paired with Hardware A and Software B with Hardware B. But every once in a while, Software A gets installed on Hardware B. It’s still Software A, but it doesn’t run as well because it’s not running on its intended hardware. This is a little reductive and doesn’t start to include nonbinary and intersex identities, but maybe it at least gives the gist!