If someone uses software that you design, it's fair to assume that they'll use it in a certain context. They might be checking their phone at the office, or quickly ordering something to eat on the way home. Whatever you design fits into a larger story about the user's experience.
The storyboarding method takes advantage of this context to help you come up with ideas. You take the scenario that you imagine your user will experience when they decide to use your software, and you draw it out as a literal storyboard, like a cartoon strip. This storyboard features the person using your software, but more importantly includes details about their experience leading up to, during, and after they use what you've designed.
Seeing this story laid out in a way that's easy to understand can help you come up with ideas about what features the software might need, and about what challenges the person might face while using your software, and that you need to keep in mind. Constraints like these challenges help creativity, since they give you boundaries to work within.
Margaret has been asked to design an app that someone can use while they're shopping to tally up the nutritional value of the food they're buying, and highlight any types of foods they need more or less of. Margaret is working on the screen where the user can point the phone's camera at the barcode of each item, and see it added to the tallies.
Margaret needs to come up with some ideas to make this screen easy to use. She pulls out a piece of paper and starts drawing a storyboard of the user's experience. It starts with the person walking into the shop and picking up a basket, then pulling out their phone and opening the app.
Immediately Margaret realises that people are going to be have to use the app one-handed, because they'll be taking items off the shelves with the other hand. She writes a note to remind herself to keep every important action within reach of the user's thumb.
Next she draws the person realising that the last item added has taken them over their sugar limit. This inspires an idea: Always show the last item added to the tally, and give the user a quick way to remove it so that they can put it back on the shelf and look for something more suitable.