Well, for me, explaining trans wasn't easy. Especially because my father for one has always been pretty Republican. I grew up with him telling me that gays and the like belong in the closet and should never come out for the good of everyone around them. Seriously, I was like eight and he was telling me this kind of stuff. But that's beside the point. Anyway, when it came to me telling him, I just told him that I was a man and that I intended on getting a sex-change one way or another. He already knew what trans was, because of the Chaz Bono stuff all over the TV, thank goodness for that.
But even now that I've been out and openly transitioning for over eight months now, I still get asked, "Wait...what's trans? Does that mean that you have both parts?" Seriously, I get that question a lot, and after I laugh my ass off, I simply reply, "No. What I mean is that I was born into the wrong skin, and since I was, I'm fixing it so that my skin matches the rest of me," I don't know if that kind of description would help you though.
Honestly, I think the description depends on the kind of person. You could have twenty people, explain to all of them at the same time, and have two thirds of them staring at you like, "Was that in English?" Think about the way that different people learn. I don't mean, draw a picture, but say...maybe the people you're dealing with are more logical and statistical, you could talk about it from a biological standpoint. Whereas more creative or non-linear people might connect better if you give them examples of your feelings and the way that it makes you feel to be perceived as a man. I don't know if any of this helps or if I'm just over-analyzing, but regardless, good luck!