I was thinking about this t'other day following an incident where in a split second I had to be there, and a split second later, I was.
It brought to mind something I was told once by my good friend Bruce, who'd been a jackaroo (aka cowboy) in Australia. He said that if you had a good horse, it would be really sensitive and responsive and quick to the reins and/or feet; but when you rode a good horse all day every day for weeks or months, you didn't need him to 'be responsive' - you & the horse were as one. The horse knew, at all times, exactly what you wanted to do and where you wanted to go, and would do it/move there instantly, without you (consciously) giving any indication. Riding my fixie is a bit like that.
It is, I think, something to do with the lack of any intermediation. In a sense, your mind is not involved. You don't have to think about gears. Or, mostly, about braking. You want to accelerate; your legs push harder. You want to slow down; they drag. No decision-making, no delay, no 'in between'; it's instant, and direct in a way no geared bike can ever be.
I'm not knocking geared bikes. Far from it. I have two, and ride them. For distances, for touring, and for timed events. But for the kind of riding I do, which is five mile urban trips for the most part, the fixie is my first choice every time. They are low maintenance, and quiet, and less likely to get nicked, but that's not why I ride fixed. I ride fixed simply because I enjoy it more. Much more. And I live on a hill!