Obviously, speaking as a medieval history/lit nut, my personal answer is easy: they’re fun and cool.
But, this answer is certainly not even quasi-scientific and probably won’t work for the larger audience that is western society.
First, a little history. For the last few decades, the traditional literary canon has been under attack, and with good reason. I definitely agree that we need to include certain “minority” authors who have been overlooked (though not all, Elizabeth Keckley, for instance, couldn’t write to save her life) and we need to make room for modern authors (hey, I’m also a sci-fi/fantasy geek). However, this inclusion doesn’t mean we should completely ignore canonical authors such as Chaucer, Homer, Virgil, Ovid, Shakespeare, Spenser, Malory, and the rest.
Unfortunately, I got through four years as an undergrad English major with no opportunity to read Beowulf, Spenser, Malory, Dante, or Milton in a literature class. My experience with Beroul and Langland came in a history class. On the other hand, I read Ann Petry and Ed McBain for classes.
This still begs the question: why should we support/preserve the canon? My relatively simple answer is that the canon is the basis of much of western society’s modern literature and culture. For instance, Tolkien couldn’t have written Lord of the Rings without Beowulf, The Song of Roland, Celtic mythology, and the Arthurian romances; Lucas couldn’t have written Star Wars without knowledge of Roman legend, Malory, and Shakespeare; many of Pratchett’s best scenes would be nonsense without his (and the reader’s) knowledge of Shakespeare, John Webster, and Arthurian romances. Without at least a passing knowledge of “canonical” works, we as readers and viewers miss many layers of the texts we read/view every day.
This influence is not the only reason to preserve knowledge of the canon (nor is it the only influence on western literature/film). At a really simple level, canonical texts such as Chaucer, the Gawain-poet, Malory, Marie de France, Homer, Virgil, and Aristophanes are just fun to read. Chaucer, Shakespeare, and Aristophanes, for instance, still make modern audiences laugh uproarously (see: “The Reeve’s Tale”, Much Ado About Nothing, and Lysistrata respectively as examples).
Additionally, these works tell us something about the periods in which they were written. Sometimes they show us that humanity hasn’t really changed all that much. Case in point, Aristophanes shows us that similar social issues were present in his day and the modern world, and that people laughed at similar events, even if we don’t get all his jokes. In this way, these pieces also tell us something about humanity/ourselves in psychological terms, or at least psychoanalytic terms. When we apply Jung, for instance, we find (through Joseph Campbell) that ancient heroes such as Aeneas and Odysseus go through the same cycle of growth and challenge as modern heroes like Conan, the Skywalkers, and Neo (even if modern heroes have worse actors playing them). This tells us something about humanity as a whole. The same goes for the other archetypal figures and events we see throughout early western literature in the canon.
Admittedly, I’m a bit biased on this, being a medievalist and focusing my research along Jungian lines, but I think the theory holds up pretty well. Constructive challenges are, as always, certainly welcome, though.