Excellent question, Stephanie, and one I have given some thought to. I can't claim to be able to give you a definitive answer, mainly because that would require me to be able to read cisgender people's minds. I'm not a mind-reader, and anyway a lot of the real hard-core trans-haters don't seem to have too much in the way of a mind that could be read. But I can give you some conclusions I've come to based on the things I've seen cisgender people say on various forums when trans issues arise.
(1) The first reason a lot of cispeople fail to understand us is that they don't want to. Like gun control and abortion, e.g., ->-bleeped-<- is a hugely emotive issue. Trans-haters don't want to think about it, don't want to understand it. They simply want to condemn it.
(2) Cisgender people seem to be exclusively "body-oriented". That is, their concept of sex/gender is based solely on what sort of genitalia a given individual has. If you've got male genitalia, you're a man. If you've got female genitalia, you're a woman. For them, the question is that simple.
When discussing ->-bleeped-<-, either among ourselves or with cispeople, we need to bear in mind that a cisperson's definition of "a man" is "a cisgender man"—that is, someone with male genitalia. A cisperson's definition of "a woman" is "a cisgender woman"—that is, someone with female genitalia. These are their concepts of "man" and "woman" because they don't have our experience. So that if we're talking to cispeople and we're using the terms "man" and "woman", they're automatically going to be thinking of "cisgender man" and "cisgender woman". They can't help it. Those are the only concepts they have. Which means that we're not going to be able to communicate with them.
They want to leave the brain/psychology/mind (whatever you want to call it) out of the equation because they don't seem to see it as relevant. It's hard to say why that is. I think they simply see the body and mind as a package deal. If you've got certain genitalia, you've got the corresponding psychology. They can't imagine it being any other way.
We want to talk about the psychology because that's how we define ourselves. They want to bring everything back to the body because that's how they define themselves.
(3) So we have a failure to communicate. I myself have been searching for some way to break the deadlock.
Cispeople fail to understand the distinction between sex and gender—that is, between body and mind. But the standard explanation of that distinction that we transpeople propose seems to me inadequate. We often define "gender" (as opposed to "sex") as your perception of yourself. If you perceive yourself as female, you're female by gender. If you perceive yourself as male, you're male by gender.
But I don't think this gets us anywhere. Because the trans-haters will simply retort, "Well, you're a man (because you've got male genitalia), but you perceive yourself as female. You're clearly deluded." Or, "You're a woman (because you've got female genitalia), but you perceive yourself as male. You're clearly deluded."
This standard explanation also strikes me as inadequate simply because I don't merely perceive myself to have a female psychology. I do in fact have a female psychology. I don't want to talk about perceptions when what we're dealing with is plain fact.
So I'm looking for different terms in which to explain ->-bleeped-<- to the uninitiated. Here's an analogy that I've come up with recently and that I find a very good explanation of how I feel about myself. Let's put it in "spiritual terms", because this is something that most people (even the non-religious like me) can get a handle on:
The body is merely a shell that houses the soul, and what is truly important about a person is the soul because that is their true self. Now somewhere out there, up in heaven perhaps, there's a host of souls waiting to be implanted in bodies in order to enjoy their time on earth. Some of them are male souls, some of them are female souls.
When a woman conceives, if that embryo has XX chromosomes, a female soul is assigned to it. If it has XY chromosomes, a male soul is assigned to it. But sometimes a mistake is made. The angel (or whoever) in charge of assigning souls to bodies nods off, and it can happen that a female soul is assigned to an XY embryo or a male soul is assigned to an XX embryo. When that happens, you have a transgender person.
So this is the way I'd explain myself to a cisgender person: I was born a female soul in a male body. And how do I know that I have a female soul, that I am a female soul? Well, because I prove it every day. I live female, I'm drawn to things female, I feel at home in the female camp, I find peace in the female camp. Let the trans-haters or anybody else disprove that notion.
With this analogy I move people away from the limited definition of sex/gender being linked solely to one's genitalia. I am a female soul, regardless of my physical configuration. You can talk about my body all you like. I'm talking about my soul. You can insist as often as you like that I was born with a male body. I've already conceded the point. You can insist as often as you like that someone with male genitalia is "a man". But I'm talking about my soul, which is female—and it is my soul that makes me transgender. A man like you, my trans-hating friend, doesn't have a female soul as I do. That is the difference between us.
Now the trans-haters won't like this analogy. That's OK, because they don't like anything. But most cispeople (at least in our part of the world) aren't trans-haters. They're merely puzzled. Perhaps this analogy that I use, or something similar to it, will give them a notion of ->-bleeped-<- that they can get a handle on.