Why would anyone ride a fixed-gear bike outside of a velodrome? Especially in the city they are impractical and even dangerous (we're talking without any brakes here). Stopping distance is a problem even at slow speeds if a pedestrian steps out in front of you, and at high speeds it's just asking for trouble. Knees pulling themselves apart is also a problem.
But the benefits & fun maybe outweigh the downsides. There is certainly the appeal of the minimal bike - it's lightweight and quiet and easier to upkeep. It's sleek and sexy. Probably the gearing is low so you tend to go slower which is safer and forces you to smell the roses a little bit. You notice momentum more, your body being directly linked to the road surface means you get a better sense of your velocity and the tiny acceleration changes caused by slopes or your own physical effort. Flow is more important than ever before, you look further ahead, and you can slow down no-handed. There's a tendency in some people which is akin to pimping out one's car whereby they fetishize their bike and the individual parts, but this has nothing to do with the ride so we won't get into it here.