A pro-life person could say, "I believe that life begins at conception (or when the heart starts, or when the brainwaves start), and after that, abortion is murder. But I also understand what a horrible position that puts many women in, and the real pain it causes in real lives. I still can't compromise - I believe abortion should be illegal - but I can understand why people are on the other side, and I don't hate them for being there."
And a pro-choice person could say, "I care about womens' ability to choose what they want for their own lives. But I can see how, if you genuinely believe that abortion is murder, you simply can't support it in good conscience. I can't agree with you, and we can't compromise. But I can respect you for standing for what you think is morally right, even when I think you're wrong on what is actually moral in this situation."
Wouldn't that be better? It's still an unresolvable issue. But if we could remove the hatred on both sides, wouldn't that be better?
This might not always be possible for disagreements, but sometimes it can help to step back and consider the issue and it's implications for society from other angles.
So even if compromise is indeed impossible, understanding where they are coming from is usually vital to beating them. Whoever "they" are.
There is also the personal growth angle, but sadly, that doesn't seem to be compelling to a significant fraction of folks.
The situation is better in that the listener has a more empathic, more thorough understanding of the speaker.
In that sense there was a positive exchange of information and understanding, which is great.
Beyond that, though, what has been accomplished?
I know that sounds hopelessly utilitarian, but in the end, when it comes to issues like, say, abortion, I fail to see how this kind of understanding is valuable unless it further leads to a coming together of minds to form agreement or compromise.
Now, you mention that "wouldn't it be easier for you to work out something if you don't think the person you are talking to is actually just crazy or unintelligent"?
My claim is that it doesn't matter. That whether or not I understand the other person, the very nature of the disagreement and the very fundamental principles upon which the two positions rest, makes compromise impossible.
After all, how do you compromise with someone who believes, run down to their toes, that abortion is equivalent to the murder of a child? You can't. That belief system doesn't make room for even entertaining the idea of compromise.
And the clever thing about modern politics and rhetoric is that they've taken issues that you wouldn't think would be wedge issues, and transformed them into wedge issues by building entire moral edifices around them.
The result is that when you debate someone about something as dull as, say, tax policy, it's no longer about debating facts and ideas, but rather debating values and culture and beliefs. And when you debate those values and beliefs, what you're actually doing is attacking the very things that define who that person is.