There are many different ways to analyze the meaning of a fable depending on what type of analysis would be useful in a given situation. People of different ages and educational focuses might also analyze a fable quite differently. The component parts of a fable often bear an interesting relationship to other fables, and analyzing groups of stories together can be very useful. It is also possible to analyze the meaning of fables purely as literature, although this is sometimes problematic for stories that do not have single authors. Generally speaking, any insight into meaning that comes from reading or thinking about a fable is a form of analysis, though it may not fall into a specific category.
One of the most common ways to analyze the meaning of a fable is to look at the fable in terms of what its lesson might be. Many fables are used as morality tales and specifically aim to teach the reader a lesson. Often, this lesson has evolved over time to be appropriate to the age in which the story is told. The meaning of a fable might be different in current readings than the original intended meaning.
For this reason, it is also possible to look at the meaning of a fable historically. Situating the fable in context both in the age in which it was told and among other tales of the same culture or author provides unique insight into possible meanings of the tale. Fables are unique in that they often represent cultural values, which means that historical information may have a direct impact on possible meanings of the tale. Even so, it is important to remember that this type of story has also often evolved over time, and looking at earlier versions of the same story may reveal divergent meanings.
Fables often include a number of literary devices that are used to construct meaning within the tale. In many cases, fables have multiple authors or change over time which can make the idea of intention somewhat complicated, but devices used without author intention still contribute to the meaning of the finished piece. Looking at how the story constructs its point internally can be an interesting way to analyze the meaning of a fable, though this type of analysis sometimes contrasts with the explicit moral of the story.
Analysis is a process of drawing insight from a text, and there are many different analytical strategies that can be applied to fables. Different schools of thought may come up with different meanings, as may different age groups. Looking at the fable from multiple angles can often help elucidate the true meaning of the story if one exists.