It means that if you refer to me using a pronoun instead of my name that you can use the pronouns from the set of pronouns I provided. Although you might have been given an abbreviated pronouns set (e.g. “ze/zir” or “he/him” or “they” or “ey/em/eirs”), every set of third person pronouns in English has five forms (though sometimes certain forms are the same). So, for example, he/him pronouns refers to he/him/his/his/himself as the five forms. Ey/em/eirs might actually refer to ey/em/eir/eirs/eirself as the five forms.
The term “neopronouns” tends to refer to pronoun sets developed from the 20th century (or sometimes 19th century) to today. Many of them are actually not that new. A few examples of these are described in greater detail on Wikipedia, where you can also find examples of how to use some of them in sentences.
Often, people make assumptions about the gender of another person based on a person’s appearance or name. Then, they apply those assumptions to the pronouns and forms of address used to refer to a person.
Whether or not these assumptions are correct, the very act of making an assumption can send a potentially harmful message -- that people have to look a certain way to demonstrate the gender that they are or are not.
If someone shares their pronouns with you, it’s meant to disrupt the culture of making assumptions, and to provide you with the information you need in order to refer to them appropriately.
Just as we generally have names we go by, we also tend to have pronouns that we want to be referred to by. The name or pronouns someone goes by do not necessarily indicate anything about the person’s gender or other identities. Names and pronouns tend to be publicly shared, because they are part of the language commonly used to refer to people. However, identities tend to be private (i.e. many people don't proactively share their gender just as many people don't proactively share their race, class, or sexuality with mere acquaintances)
Although neopronouns tend to be gender neutral or might be specifically meant to indicate a transgender or nonbinary person, a person who goes by neopronouns could actually be a man, a woman, both, neither, or something else entirely. Again, because people's genders tend to be private, the sharing of pronouns should not be taken as an invitation to ask for potentially private information about someone's gender.
Please note that some people go by multiple sets of pronouns or by certain sets of pronouns among only certain audiences (just as some people have different legal names from the primary names they go by, or special names they use in certain spaces, such as pen names or performer/artist names). If in doubt, ask.
You can also share your own pronouns by sharing a link to the pronoun you go by. Here are some of the more common ones:
This website also provides much more explanation, examples, and information about pronouns and gender inclusive language. See the appropriate section for more: