My particular collection is distinguished by three colors:
Black is the original set that my aunt showed me. They don't have a particular theme.
Blue is all about actions and movements.
Green is all about "Voyages". I'm not sure what that means but its got cool looking treasure on the faces.
Now before I can get into making backgrounds with these cubes, a prior knowledge of the Heroic Journey is required.
Basically each set of story cubes corresponds with the first three steps in the Hero's Journey.
The first step is the Call to Adventure. This is the person, event, or thing that spurred the character to leave his home in the first place to go adventuring. It could be a skeleton in the closet fell out; an urgent letter; or a panicking cousin pleading for aid. I use the black set for this step.
The second step is the Refusal to the Call. I interpret this step as to why the character can't go back to his home and ordinary life. It could be debtors hounding his back; a passion killing; or some other event not related to crime. I use the blue set for this step.
The third and final step is Supernatural Aid. This is something or someone that aids the character on his journey. However its purpose doesn't have to be immediately obvious. It could be a lucky charm; an old neighbor friend; or the blessing of a long forgotten god. I use the green set for this.
And that's basically it. Roll three dice of different colors and divine away!
Here's a few examples to illustrate how this mechanic works in action:
As you were about your daily life a giant tortoise approached you with a mushroom in its mouth. It plopped it at your feet and to your surprise began to speak to you. It warned you of the coming curse and said the second ingredient for the cure rested in a far off land. Then you awoke from the dream. You discovered that the mushroom was in your hand and that your entire village was trapped in an eternal sleep.
One day a magician visited your village with a cart full of delicious looking food in tow. He offered the delectables to you all. However shortly after everyone's consumption you discovered that you all had been struck blind. With a tut the magician said, "I must have missed something crucial," and scribbled something. He placed a pair of spectacles on your nose and you vision was miraculously restored. "Unfortunately that was and will be the only pair in existence," he told you, "Come! I'll need your assistance to fix this mess and perfect my divine delicacies!"
You received a letter. The letter warned you that you would murder your neighbor and try to hide the body. Dismissing it as slander, you threw it away. A week later you awoke in your bedchamber covered in mud and blood with a shovel across your chest. A raven perched on your window seal cawed "Follow! Follow!" before fluttering away into the distance.
Like with any divination the meaning behind these symbols is in the eye of the beholder. Let yourself and the player interpret the cubes and together pick what sounds the most interesting or convenient for your game.