Disclaimer: This is how -I- do it. This may or may not work for you. If you find a better way, then that's awesome for you.
Warning: This is mostly focused on therians. I added a part on more general otherkin, but I don't actually have experience with that part, so I'm playing it by ear and my interest in psychology.
First of all, remember you don't have to tell anyone. It's something personal and if it doesn't affect them (which it shouldn't unless you're very unstable, then you should probably seek a therapist anyway), then why tell them?
If you decide to tell someone, the big thing is not to be really crazed about it and don't overthink it.
Don't use huge words (like therianthropy or otherkin) that won't be easily explained (try explaining the entirety of those terms in a single sentences, I dare you).
Don't make a big deal out of it. Because, guess what, it isn't a big deal. It's something very personal. It's a big thing to you because you live it, but outside of you, it doesn't actually mean much of anything to your normal human life.
I always like to bring up reincarnation before I say anything (if you don't believe in reincarnation, then you can always do something else to test the waters). I start with my own beliefs about reincarnation and I edge into the idea that people can be reincarnated from animals, and usually by then I can definitely tell if someone's going to be open to the idea or not, because if they're not they'll be totally rejecting of the idea. If they are open to it, I've have people jump ahead and go "well you must be reincarnated from a cat then!" and when I agree, we laugh and that's around it.
They don't have to go into it any more than that. Being reincarnated from another animal, or having the spirit of another animal is usually the best place to leave the discussion. Otherwise you can start sounding crazy and like you need mental help.
(If you are otherkin and decidedly not therian, then you might have a little harder of a time. I find that people are more likely to believe therians because it's a fairly simple idea ("I have spirit of [insert Earth animal]" rather than "I have the spirit of [insert being that you might or might not have any way or describing in simple English]"), and one that's talked about in a few of the popular religions. Being an elf or a dragon (I'd have to say, telling people about angelkin is always the worst) makes it more difficult because you're not on level grounds with people all the time with what they believe that mythical being to be.
I find bringing up the idea of other worlds within the realm of reincarnation can be useful, as well as the idea of calling something, say a 'dragon', because it's green huge and has scales... but it might be a dinosaur or another creature, but the name 'dragon' is easiest because it gives the general idea of the creature without having to say "a huge green-scaled being, that may or may not have feathered wings and bull horns".)
Note: (If these topics are too philosophized for your friend, it might be easier to just go for the kill, without being too serious: "Hey, I have the spirit of a dragon!" "And I'm the Queen of England, now can we just go sit down?" If it doesn't go well, just pass it off as a joke and drop the subject. Remember, they don't have to know.)
If they ask, or the topic reappears, I'll sometimes go more into it. There's only a couple people, whom I trust not to freak out, that I've discussed phantom limbs with. Generally the therian community never gets brought up, unless the friend is also a therian. Things like m-shifts and ph-shifts, I don't really talk to anyone but other therians about, because it just sounds far too strange to people who don't experience it.
That's basically my views on telling people. I won't lie and say I haven't made mistakes in judgment and had people call me crazy--but if they're good friends, when you drop the subject, they'll drop the subject (or continue calling you 'the crazy cat lady' as a fond joke). Some of those friends even eventually grew to accept it. Others don't even remember my telling them at all, and that's probably for the best.