It's probably easiest to understand Prof. Williams' work if you imagine an acoustic performance by musicians on a stage in front of you. The farthest-left and farthest-right sound sources are some distance apart from each other, so imagine a straight line connecting them. Then draw straight lines from those two points to your recording position. You now have a triangle with yourself at one of its angles--and you're facing what we hope is a pretty-much-symmetrical left-to-right arrangement of sound sources.
Now consider, as a parallel situation, your home stereo system. Assuming that you have two loudspeakers some distance apart, and you sit or stand (or lie, or kneel, or dangle from a chandelier) at one corner of a triangle as you face those speakers. The assumption is that this triangle will be approximately equilateral. And the assumption is that during playback, you want the relative positions of the original sound sources to seem to originate from places in or between the two speakers, corresponding to the relative positions that they actually occupied during the performance.
If the angle that you faced in the hall was 60 degrees, for example, then the geometrical correspondence between performance and playback will be 1:1, if you have achieved this ideal. If the original performers occupied a much wider angle with respect to your recording position, then that wider angle will be squeezed (proportionately) down to the 60 degrees that you have during playback. (If the original performance was much narrower, that's a bit of a special case, because in general it sounds very unnatural to stretch the angle by much--you get "the violin that ate Manhattan" effect which can be, um, rather disconcerting ...)
All of his charts and graphs and magical incantations are aimed at (a) getting the stereo image to occupy the intended width in the space between your speakers, including the speaker positions themselves as endpoints if that's what you want, and (b) keeping those angles proportional--i.e. whatever sound source was in the center, if it was really only as wide as whatever was on (say) the far left, it shouldn't play back sounding as if it was much wider or much narrower than whatever was on the far left. "Everything in proportion" (geometrically speaking) is the idea.
He is also very much concerned with the pickup of reverberant sound, both quantitatively and qualitatively. But that's maybe too much to go into here; I just wanted to set out the basics.
(By the way, I've known him for years from numerous meetings at AES conventions. He's a quick, funny, warm-hearted guy with twinkly eyes, who seems to be well liked and respected by those familiar with his work. And by no means is he the stereotype of a theoretician who only juggles numbers and equations and doesn't listen; he does a large amount of live recording and is constantly tinkering and experimenting with different recording and playback arrangements. Last I heard, he was heavily into 7.1-channel systems, for example.)
--best regards