With Keys of the Past you can complete a stage of events that aren't available on the day you use them. Even if you haven't completed the stage beforehand (although you still need to complete stage 1 before you can access stage 2 etc). If the event is already available that day it won't show up in the list.
Instead of using stamina you will use Keys of the Past to complete the stage. Depending on the difficulty the stage might require more Keys of the Past.