Not many people are aware of these, or maybe you are and you just don’t fully understand them. Essentially, neopronouns are new pronouns people have made to better fit their gender no matter what that gender is. Many people who fall under non-binary or nonconforming genders tend to use them, though not all of them do. Of course, I cannot speak for everyone as to why they have chosen to go by a neopronoun, but I can only assume it is because no other current pronoun fully fits who they feel they are. Or, perhaps there is an aspect of the odd resistance to using they/them as a singular pronoun for someone (despite people doing so on a regular basis all the time), so they may argue it is easier using a neopronoun than it is trying to get someone to use they/them. The person may also like the idea of using less human or binary pronouns but do not want to go by it/its (though some people do), which neopronouns are very helpful in that case as well.
No matter what the idea, it really comes down to the fact that people use neopronouns because it helps them feel more comfortable and like themselves. It is really no different from a cis-female using she/her, a trans male using he/him, or a non-binary individual using them/them.
Some examples of neopronouns are:
Fae/faer
ae/aer
e/em
ey/em
xe/xem
ze/zir
ze/hir
ve/vir
Some of those pronouns, such as e/em and ey/em are sometimes referred to as Spivak pronouns, which fall under a range of pronouns used in some way by mathematician Michael Spivak. While he himself did not create all of them, with many spanning from different people that goes back to 1890 by editor James Rogers, the different pronouns can sometimes find themselves gathered under the Spivak label.
Another type of neopronoun example is called a noun-self pronoun. These, as stated in the name, tend to have a noun of some kind before attaching -self at the end of them, though sometimes the individual will just go by the noun. Common ones of this may be bunself or faeself.
Sadly, there are a good number of people in the world who do not want to use neopronouns. This can be for a variety of reasons. Maybe they don’t believe in more than two genders, maybe it is simply because they do not understand, or maybe it is because they view these pronouns as random words or sounds that don’t have a real meaning and therefore should not be taken seriously. In fact, there are some even in LGBT+ communities that do not fully accept the presence of neopronouns because they are worried it will make the community seem like a joke. Now, I can understand the fear, afterall we have been fighting for recognition and the right to be ourselves for a long time only to be called horrible things in return. However, the use of a neopronoun is not something that is being done as a joke, at least not for the people who legitimately see neopronouns as the best way of expressing their true selves.
But think of them this way. Just like all words, if used in the correct context and often enough, these could easily become part of daily vocabulary. If taken seriously and thought about appropriately, they can have a place in a person’s life. Pronouns like xe/xem are no more different than she/her in origin. Both were created to help with talking about the individual, except one is older and more widespread and accepted than the other. And yes, they may need some getting used to, and people will need to be aware of what neopronouns they want to use (that is making sure you are actually being respectful to yourself and other cultures, as well as watching the noun you may choose for your noun-self pronoun), but otherwise it is just an adjustment to the language.
And really, as long as you are not causing any harm by either the word in use (in the case of the noun-self pronouns) or the pronoun you are using, then what is the problem with referring to someone with their preferred neopronoun?