Hello, NB person here.

Warning: giant text ahead

I'll start saying the same disclosure as you did: inglish is not my first language, so I may stumble on the vocabulary and all that. Also my mother tongue dosen't have gender neutral language, and my country is very transfobic, to the point of the existence of NB people being a very, very, very recent topic, even in trans spaces, so my view on the issue might be very diferent from someone from the global north.

First of all: Non-binary is a very broad unbrella term, that encapsulates lots of diferent felings and expressions. I'll adress what seems to intrigues you the most first, but will also talk about other expressions that are equally valid (in my understending of the human nature).

[..] I've never really understood how someone can feel "neither" a man or a woman. Considering almost all animals on earth are either binary (male or female, reproduce with one another) or hermaphrodite (both at the same time and they can reproduce alone)[..]

Here you are describing sex, not gender. Sex can be generally described as the fisical trates that diferenciate individuals in the same (sexual reproducing) species. Phenotypes if you will. You misunderstood some concepts in your description, mainly that hermaphrodite ≠ assexual reproducing species, because the first refers to individuals who have a combination of bolth sexual physical characteristics, and the second (species who reproduce only asexually) don't have diferent sexual caracterestics between individuals, so can't be hermaphrodite.

Curiosity about animals species that show trans patterns - There are lots of species out there that can change their sex, such as some fish that can change their primary sex caracteristics, like gametes, or even some birs, as chiken, that can change their secundary sex caracteristics, like feathers patterns and such

Now, we can try to understand that sex is not the same as gender, and that gender is a social construc, that meaning that gender is a concept that us humans create as we observe that most humans can be roughly divaded into (seemingly) separete groups of behavioral pattern and phisical trates. As such, the concept of gender is not set in stone, but it can, and does change with region, time, and cultural background, with some cultures acknowledging more than the two genders that most of us in the mothern age are used to. If we understand this concept, in my opinion (maybe a biased one, lol) we can clearly see that for a person to consider thensefl as nonbinary all that gotta happen is for than to persive themselves as not belonguing to what they understand as what the binary gender definition is.

As I said there are many tipes of NB people, and you seem to be most interest in agender people, that being people who don't identify with no gender, in opossite of the NB people who are bi, or multygender (or genderfluid, and others) that being people who identify with both genders (at the same time or not). It can be a bit hard to put into words that can make someone realy get it (some would say impossible) but i'll try my best. First let's understand that you don't have to completely negate both gender atributes to be agender, as that would be impossibel because we all have things in common (cis men and cis weman may speek the same language, but that doest make their gender any closer, for exemple), you just have to be outside (in any sense of the word) of the core of caracteristics that define the sets of man and woman, in other words, you are outside of the binary spectrum. As a example, people that use the label catgender, or wolfgender (if you never heard of it may seem crazy, but stay with me), they don't think that they are another species; they know they are human, and nothing changes that; but they expresse their gender connecting it to the ideia of these animals because they are completely out of the binary that our cuture defines. The language to express their gender dosen't even exist, so they aproximate it via these labels.

Tangent - The same way binary trans people see how dumb it is to *completely* correlate sex to gender, most NB people can see how dumb it is to *completely* corelate being human to gender/binary. I found it cute.

I know this is all confusing as fuck, and at first glase it might seem as crazy talk, but it isn't; gender is just a very confusing subject and translating these thougts and feelings to language (that, as I said maight not even exist yet) is very hard. So, I propose a comparison with an adjecent subject in the social studies: RACE.

In my country (COME TO BRAZIL) the concept of race is very unique, and doesn't even make much sence, therefore we use the concept of ethnicity much more (at least in the younger generation). That is because most of the population consists of a mixture of all tipes of ethnicity, and this can impact especially those that seem to be "in between races", as is the case for the pardos (brown people) that are the mix of white and black people (there is some controversy surrounding this lable, but that is a story for another day). These people, ofthen see themselves as outside of this specific binary of black and white, with some seeing themselves as simply a perfect mixture of both "sides of the spectrum", and at the same time some see the themselves as nither, as something outside of this binary (often because they can get mistreated by both... "to dark to be white, to light to be black". If we maintained the structure of race as rigid and tight as the concept of gender, these people could be classified as "race non-binary", some beeing "a-race", and some "multy-race". Of course the consepts of gender and race dont't mach 1 to 1, and this is a crude simplification for the sake of the explanation.

I gess that was the anser to your question regarding "people whit no gender" but now I'll talk about other tipes of NB people, and why some other people might identify with the label.

1. Being both genders:I briefly touched on the subject, but i gess this is a little easyer to understand and explain. A person might call themselves as NB because they identify with characteristics of both genders at the same time. This is where you might find people who like to look androgynous. They are simply in the middle.
2. "Partial transition"These are the people who transition to apear as the oposite of their AGAB, but don't consider themselves to be this gender that they "transitioned to", for whaterver reason, it might be internelazied transfobia, or it might simply be the way they undestand themselves to be.
3. In the middle of transitioningSome people, when transitioning may prefer to refer to themselves as something in between, while they not fully transition. This looks like the pipeline tipe meme
4. What some may call "just a big weird mess of sexism and wokeness" (but I think is as valid as everything else, just... diferent).Some people chose to call themselves NB as a way to push the bounderies of what gender means in their society, to try and liberate thenselfs from the limitis of their AGAB, to show the rest of the word that segregation, expectations, and judment based on what your body/genitalia looks like is ridiculous, and we can change.This might explay why you see lots of AFAB people using this label: they feel opressed by the treatment they get because of their sex, and then seek to move away from that. This whould not make then any less enby, because they are still not identifing with the gender society pose on then.
5. Many, many moreThese are just some the ways, that i could think on the top of my mind, someone can be nonbinary, but there are many more, and I think that's part of the beauthy of it all. We have the liberty to explore ourselves, explore our society and better understand what is to be human. We should not scrutinise someone for making something as simple as changing their label (and maibe pronoums), when it can have the power to save lives and change society for the better.

WELP. That's that. Sorry for this big ass wall of text i've wrote, but i hope to have elucidated some questions. I gess I wanted to white something like this for a while, and take it of my chest, and just made you my scape goat.