The word didactic is an adjective referring to something with an inherently instructive function. If a story is described as being didactic, then the story's express and explicit purpose is to teach. The operative words here are express and explicit, and this is where didactic tales depart from fairy and folk tales. Fables are a type of didactic tale in which, again, the moral is expressly stated. Fables generally involve some sort of magic or talking animals, whereas a didactic tale in general isn't bound by any formal or atmospheric expectations; it is only bound to contain an explicitly stated lesson.
In the case of "The Rat Trap," Lagerlöf blends a realist approach to literature with the conventions and style of didactic tales. It is often observed that "The Rat Trap" sounds much like a fairy tale in its language, particularly beginning with the old cliché, "Once upon a time." At the same time, "The Rat Trap" doesn't offer its readers any comforting, magical deus ex machina. The transformation undergone by the vagabond is an interior, emotional transformation. He is still destitute. He must still scrounge for his living. He just has a different outlook about it. Lagerlöf also turns her eye toward the changing economic landscape of Sweden in the face of industrialization, and how that shift leaves some people to face new economic challenges.
Lagerlöf's story qualifies as didactic only because of the vagabond's note at the end of the story. Surely, the image of the kronor in the rat trap would be enough for the reader to process the vagabond's redemption arc. But Lagerlöf takes it a step further, and in traditional didactic fashion, leaves nothing to the imagination. She writes out the vagabond's whole note to Edla, which concludes, "the rat trap is a Christmas present from a rat who would have been caught in this world’s rat trap if he had not been raised to captain, because in that way he got power to clear himself" (74). Although the language remains in the figurative realm, the clear explanation of the moral, particularly in the final clause, qualifies the tale as didactic. While to a modern audience the explicit statement of a story's purpose may seem like an egregious literary faux pas, it was necessary for the writers of morality tales to state their purpose so that it would be impossible to misinterpret their story to promote values that conflicted with their own.
