RE: How Do you Explain Therianthropy as not "weird?"
In short, you can't. How someone reacts to being introduced to a new concept is not something you can control. There's some good advice here, though, on how to stack things in your favor, and I do intend to contribute to that question later, but first...
I'd add that there are very few people in our lives that need to know, though - so the absolute best way to avoid someone having a negative response and thinking therianthropy is weird is... to never tell them. I think anyone looking to share their therianthropy with the people around them need to ask themselves at least one major question - "Why do I want to tell them?" Note that this is not the equivalent of, "Why do they need to know?" An internal dialogue that tries to find justification for something by addressing it as the "needs" of others is a thinly veiled self-serving delusion, so skip it and get to the real point - why do "you" (the general you, who is reading this now and contemplating telling someone about your therianthropy) want to tell "them"?
If the thought of telling someone makes you happy and excited, take that as an internal warning sign that you're sharing for the wrong reasons. Therianthropy is completely intangible to anyone except the person experiencing it. It doesn't grant us special insights, unique powers, or any kind of advantage in life, whatsoever. Think of it from your audience's point of view.
Furthermore, that brings up my next point - know your audience. If the person you're wanting to tell is a religious fundamental-type, chances are, the conversation is not going to have the wonderful outcome you think it will. Different people are going to react in different ways, and the way in which you explain things can also influence how they process that information. To pick on the religious fundamentals again (don't worry, I'm Christian - I'm authorized to single them out), explaining therianthropy by relating it to gender identity is not going to work in your favor. You know the person you want to tell better than anybody on these forums ever will, so just remember that the comments you see here are generalities that may or may not be helpful in your specific situation.
Finally, and here's where I want to try and actually answer your initial question - I'd recommend explaining it to yourself, in your own mind, over and over and over again. Play the Devil's Advocate against yourself and ask yourself questions, and be harsh and critical of your answers. Be able to explain things generally at first, and if your real life audience isn't interested in continuing the discussion further, you can stop there. Be able to explain things in great detail if you're questioned.
Remember that you're not required to use the phrases "I identify as..." or "I believe I'm a..." I think, for most of us, it would be safe to say that we want to emulate our theriotypes as much as possible. If the opportunity presented itself organically, you could start by casually dismissing the fact that you are undeniably human (for example, I don't sleep through the winter, I don't maul children, etc) but you do admire [insert your theriotype here], and seek to mimic them in a way that integrates smoothly with your (obviously) human life. You and I know there's much more to it, but again, this plays into "knowing your audience" and allows you to provide a "nuanced" explanation that can open the door for further conversation, if you decide you really do want to tell someone but fear there may be a negative reaction.
It would be fitting, I think, if among the last man made tracks on earth would be found the huge footprints of the great brown bear.
― Earl Fleming