Board Thread:Questions and Answers/@comment-25641028-20141108024644/@comment-24577221-20141108031538

That button appears when you've selected a structure or object that's in the process of doing something. Examples would be a rock or a tree that is being removed, a bakery that's working on creating some food, or the nursery while it's holding a monster egg that hasn't hatched yet. (All of these things will be showing a progress bar.)  When you click on the "Speed up" button, you're given the opportunity to complete the task immediately, at a cost of 1 Diamond for each remaining hour or part thereof.