In his book, The Grammar of Fantasy, noted writer of Italian children’s literature Gianni Rodari gives an example of re-creating a familiar fairy tale by breaking it down to its most basic elements and “recasting” the story. For example, Character A (Cinderella) lives with Characters B, C, and D (Stepmother and Stepsisters), who are invited to Event (the Ball) and leave Character A behind, etc. etc. Rodari, of course, goes into far more detail in his book, but I will leave it to you to seek his exact words for yourself.
In the meantime, and especially if you are interested in retelling a classic fairy tale, you can apply this process in greater or lesser detail (depending on what you want to accomplish) to an existing story and “recast” it with your own characters and situations to create a new one.