Yes, social intelligence is the core of it, but a lot of the time it's not so much a matter of innate ability than of mindset and values. For a long while I had this strange idea that the overly-literal way of discussing, which assumed the form but not the function of rationality, was the right way of discussing, and the huge majority of people were just doing it wrong, and needed to be informed of the rules of ~rational~ debates. Looking back, I might have been rather frustrating to talk to. :)
But then I changed my mind and reframed debate as an action undertaken by some agents with primary motivations other than debating for the hell of it, like voices in the dark. And started to look at the motivations behind what has been said, with the goal of having the most accurate model possible of how and why the other person actually generates those ideas. The chains of reasoning that output the goals the other person wants out of the debate are not terribly hard to go through, even for a self-identified social "extraterrestrial".
Those particular debate tactics aren't, however, the best idea for improving the dialogue. I've had them used on me, or rather against me, even here on LW, mostly for disingenuous purposes. They're very easy to corrupt. In particular, "so you're saying that..." seem to be the code words that herald the arrival of the Straw Man. Okay, I get it, the other person might be genuinely trying to understand your alien (to them) mind. However, when there's disagreement, in trying to assemble a coherent mental picture of you, they choose, out of the traits you could possibly have, the ones that depict you as farther away from where they come, rather than closer to that. The bottom line seems to be "... so that's why I disagree with you, because I'm too reasonable to believe all the crazy things implied by your expressed beliefs". If you're looking for feedback, it may be best if you ask rather than propose interpretations.
As well as that, telling people what their emotional states seem to be is more often than not a bit offensive. For some reason people like to come across as less emotional and more cool-headed in debates rather than the opposite. The natural reaction would be mildly defensive ("Who, me? Nah, I'm totally chill"). Even if they might be feeling a certain way, emotions aren't often consciously credited as origins of one's behavioral output. If you're using this, then a "buffer" statement might be needed, to reassure the listener that you're not considering their thoughts less legitimate because they have emotional sources, or their emotions less legitimate because they're intense ("of course, I see why you might be happy/angry/passionate about this, and that's totally okay").
The core is empathy... you have to show that you genuinely accept and make space for the other persons idea and emotions, it's the mindset behind the strategy of active-listening. If you simply use the strategy of "so you're saying that" without genuinely trying to understand the other person, it comes off as straw-manning. Similarly, if you are telling someone else what their emotions are, instead of making space for their emotion, it comes off as judgmental.
I think something else you're getting at here is that this is something that's very har...
Edit: I didn't realize this before writing the post, but what I'm referring to is The Principle of Charity.
Story
I was confused about Node Modules, so I did a bunch of research to figure out how they work. Explaining things helps me to understand them, and I figured that others might benefit from my explanation, so I wrote a blog post about them. However, I'm inexperienced and still unsure of exactly what's going on, so I started the blog post off with a disclaimer:
My friend said that it's a bad idea to do that. He said:
I interpreted what he said literally and basically responded by saying:
This was stupid of me. He didn't mean "claim that you're 100% sure of what you've written". He didn't mean "pretend that you're way more confident in what you've written than what you really are". He meant, "I think that it comes across as you being less confident than you actually are. And so I think you should reword it to better communicate your confidence."
I shouldn't have interpreted what he said so literally. I should have thought about and responded to what I thought he meant to say. (Although, he also should have been more precise...)
Thesis
People often interpret and respond to statements literally. Instead of doing this, it's often useful to think about and respond to what the other person probably meant.
For example, "If I interpret what you said literally, then A. But you probably meant X, so B. If you meant Y, then C."
Depending on how confident you are in your interpretation, you should probably respond to a variety of possibilities. Like if you're < 80% sure that you know what they meant, you should probably respond to possibilities that have at least a 5% chance of being what they meant. I'm not sure whether 80 and 5 are the right numbers, but hopefully it communicates the point.
Why don't people do this?
I see two likely reasons:
Practical considerations