It depends. Some people will say not to tell them cause they may react badly. My mom was fine with it. I explained what otherkin was, then told her I was a catgirl. Then I had to explain what a catgirl was. If you think they will react badly, its prolly a good idea not to tell them. If you think they will react positively I would start by explaining what it means to be otherkin. Explain what it is first. Then if they're okay with that, explain that you identify as an animal and tell them what animal or animals you identify as.
The general rule is to first, always ask oneself if it is worth risking a relationship with someone, especially those persons who have major influence or impact on and in life. This is parents, spouses or partners, siblings, or even one's own children when they are mature enough to appreciate what they are told and to take it seriously. The reason for this is because if one presents it the wrong way and or to the wrong person in their life who does not quite have that kind of understanding, it can sever that relationship for good or at the very least heavily damage it. So do be absolutely certain this is something that needs to be done, because if it is not, it generally is not worth risking it.
The easiest methods tend to be exploring conversations of spirituality with people, especially the idea of reincarnation and or spirit animals, often called "power animals" or "totem animals" despite those all being distinct things. People tend to be appreciative and understanding of reincarnation as a concept and it is very mainstream, especially if one enters into a dialogue about it and how they feel like they may have been some things in the past. Talking about it that way and how one feels they were maybe an animal in the karmic wheel of birth and rebirth and how it carries over. How one feels there is still echoes or bits of this past. Obviously the more one understands these religious and spiritual concepts the easier this comes; the same goes for spirit animals, that one feels they have this animal spirit at the core of their soul they rediscovered after repressing it.
"Otherkin" and "therian" are terms that need to be used careful, because unfortunately they have developed a stigma that really is hardly the fault of those who actually have those experiences. All too often are the words and ideas the butt of bad jokes. So if one wishes to use them, explain the relationship carefully and clearly, dispel misunderstandings with responses that address likely concerns. My recommendation would be to, first, give a personal explanation not the experience before and any point bringing up those terms. If those terms do not even need to be brought up, make no effort to do so.
Knowing and reading the conversation is key, as is being able to explain what the experience is like and to make it familiar to normal people who can accept it.
Ah, we've discussed this question a lot with great results. I think I should write an article about it when I find the time.
If I remember correctly, many people agree that first of all, you should get clear about why you would like to come out as otherkin. There might be good reasons, but in many cases, they are not really strict; even if you've e.g. displayed some unusual behaviour because of your theriotype, you could explain it by pretending to have roleplayed. Oftentimes, telling others does also not have a too direct advantage but may rather create problems. After all, what's told once may not easily be taken back. So, what do you intend to achieve by telling them? Is it likely that you will achieve it, and how much a risk is there?
In order to realistically assess this, I think it's vital to get to know the people you want to tell. What is their general stance on religious or spiritual beliefs, the things @Red-in-Tooth mentioned? How open are they to unknown and highly unconventional ideas? How open are they to non-classic identities, e.g. gender identities? Remember that claiming to be non-human might appear outrageous to some. E.g. if the people you're telling are strong christians, then they might think that being human is a gift given by god; hence claiming to be non-human would be pure blasphemy. Carefully checking the tide by just casually talking about religious, spiritual or philosophical topics will give you an idea about the risk you're taking and - in the best case - a starting point for coming out. Reincarnation, previous lifes, or the idea of the multiverse might be good starting points. Likewise if you're a psychological otherkin, then talking about psychology would be a good start. I think also the furry fandom is. I recently came out to a colleague of mine using the furry fandom as starting point, which worked pretty well.
If you found some common ground to start from and finally decided to give it a go, it might still be good to not just burst out that you're otherkin yourself. You could e.g. just ask "can you imagine there are people who don't identify fully human?" and discuss that a bit. This will give you the chance to still stop the talking without any harm if things go wrong. If you finally talk about yourself, be sure to make clear that this is not a game or joke, but a very sincere and honest confession of your inner self. Try to tell what it means to you, and how your own experiences feel like, in your own words. I think it's also OK to show that it's very hard to talk about it, and that you yourself don't have all the answers. Show that you're rational about your identity. Stay calm and explain any questions that arise, and show understanding if your peer is confused, surprised, or even outraged. Maybe mention that you don't primarily expect people to believe you, but rather ask for tolerance and acceptance. Admit to realize that being otherkin is something that is hard to understand if you don't experience it yourself. Also clearly state that it's not a mental illness, that you don't seek for help (unless you do), and that you're not the only one feeling like this. Remember that we are thousands. Finally, let people know your wishes about what to do with the knowledge; should they keep it secret? Should they ask questions? Should they just think about it and come back to you later?
You see that this is not at all trivial and needs careful preparation, as well as a high level of self esteem. No matter how you decide, I wish you all the best!