But the moment they encounter someone who genuinely accepts them as they are, without any judgment, they suddenly feel so much better that they can't help but want to be around you.
Yes yes yes, a million times yes. This is so true for me. My (successful) attempts to modify myself to be more social were sparked off by meeting just such a person. It was a girl I met on the street three years ago. We started talking, then went to her place and spent the night talking. There was no sexual tension at all (though we did have sex much later), I just sat there thinking "holy crap, I've been sitting in a box my whole life, I have to learn this." It was absolutely glorious to feel not judged in the least. I have since learned to project a similar vibe when I try really hard.
Followup to: Do you have High-Functioning Asperger's Syndrome?
LW reader Madbadger uses the metaphor of a GPU and a CPU in a desktop system to think about people with Asperger's Syndrome: general intelligence is like a CPU, being universal but only mediocre at any particular task, whereas the "social coprocessor" brainware in a Neurotypical brain is like a GPU: highly specialized but great at what it does. Neurotypical people are like computers with measly Pentium IV processors, but expensive Radeon HD 4890 GPUs. A High-functioning AS person is an Intel Core i7 Extreme Edition - with on-board graphics!
This analogy also covers the spectrum view of social/empathic abilities, you can think about having a weaker social coprocessor than average if you have some of the tendencies of AS but not others. You can even think of your score on the AQ Test as being like the Tom's Hardware Rating of your Coprocessor. (Lower numbers are better!).
If you lack that powerful social coprocessor, what can you do? Well, you'll have to run your social interactions "in software", i.e. explicitly reason through the complex human social game that most people play without ever really understanding. There are several tricks that a High-functioning AS person can use in this situation: