When a water droplet hits a puddle it pushes water to the side. This water then pushes water to the side of it and so on. Since the speed at which the molecules can collide is roughly the same on all sides, the shape of the wave is circular. When the water displaced by the water droplet falls back into the space left from the impact, the water nearby falls into the space created in front of it, making a circular dimple inside the main wave. The water molecules in the middle bounce back when they collide to make another wave inside. This continues until all the energy from the impact is absorbed.

The question says "freeze dried". I doubt you mean actually dried, so that there's no ice left? I take it to mean that the concentric rings are forming a pattern in the ice.

My best guess is that the puddle has frozen in stages, often over several nights, causing a freeze-thaw-freeze pattern in the ice. The other thing you often see in cold weather is ice over an "empty" puddle, where the surface has frozen and the water underneath has drained into the ground. If this happens slowly over successive nights you get some strange effects where the puddle gets frozen into a sort of amphitheatre shape.