Mostly it has to do with how other people treat you.

If we treated everyone as if their biological sex didn't determine anything about them other than who they are capable of reproducing with, the idea of "not feeling like the sex you were born with" would seem a lot stranger. Most of what people describe when talking about what it feels like to identify as an gender has to do with the expectations, prejudgement, and treatment from others.

It's really a way of navigating and managing other people's expectations for you in a world where many, many people still use sex as a way of categorizing people for things OTHER than procreation, and use gender presentation as a proxy for sex.

Choice.

Really, that's the crux of it. A person can choose to identify however they want, there's no set criteria for what would make a person choose to be identified as male or female, it's simply "What makes that person feel more comfortable?"

Now, inevitably this is confusing for people at times, especially when the outward signs would suggest that the person should be considered one or the other based on appearance. But making mistakes is part of life, and if that person tells you they prefer another pronoun, it's just common courtesy to do so.

Which one “feels” right. As a girl in online gaming spaces ~2006 I got misgendered a lot and it never failed to bother me, because it was wrong. For most of us our gender identity happens to match the psychical body we have, so it’s not usually a problem. For some people it doesn’t, and so it feels wrong to them to be referred to as that gender. (There’s also some people who don’t seem to have any strong gender identity at all, so they generally just go with what society says they are).

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As social animals, we largely define our sense of self, our identity, in relation to others. This can be inclusive, "I'm similar to these people", or exclusive, "I'm not like those people". For most of us, our individual identity becomes an amalgam of all the different subgroups we identify with.

Now, pretty much the earliest group we get sorted into and identified by is based on our biological sex, which is normally determined/assigned at birth using our external genitalia. Once that happens, we start getting socialized to think of ourselves as belonging to one group or the other and how to present that identity in socially acceptable ways. For most of us, that works fine. For some, they feel like there is a mismatch between their physical characteristics (sex) and the social expression of that sex (gender) and their internal sense of which group they should belong to.

Ultimately, there is no objective, measurable criteria by which to judge a person's identity, because it is in their own mind. We can only look at what they say and do to determine how "legitimate" their identity claims are.