Evaporation is more at the edge. More of the water makes its way to the edge. The water carries more color to the edge. So that is why the ring of coffee color is formed.
But why is the water making its way to the edge all the time?
Gravity / water pressure. Consider an overly simplified case[0]: A molecule "disappears" from the edge, leaving a cavity (blue circle). Waiting to flow into the cavity are two molecules, one on the inner side (red) and another on the outer side (purple) of the cavity. Molecule on the inner side is being "pushed" into the cavity by a much larger "body" of water (pink) than is the molecule on the outer side (light purple). So even though both molecules will move into the cavity, the inner molecule will move farther. Repeat a few quintillion times, and you've got directional flow from the middle to the edge.
Because the drop/puddle is trying to keep its shape. I think that's what the current top comment is saying about the contact line being fixed. On rough surfaces the edge can't just retreat as it evaporates, and if I understand correctly it also wants to keep the rounded shape at the edge due to surface tension, so water gets pulled in from the rest of the puddle to fill it out.
But why is the water making its way to the edge all the time?