What do swallows have to do with Arthurian legend? Well, nothing…but that's the point.
The first guard Arthur meets is very interested about how Arthur and Patsy came across a coconut—this is a fair question, because coconuts don't grow in dreary ol' Britain. Arthur's eventual response is that it could have been carried by a migrating swallow traveling from a tropical zone to a temperate one. Thus ensues the lengthy and famous discussion about whether or not a swallow would be able to carry a coconut.
That's not the last of the swallows; there are references to them throughout the movie. For instance, at the beginning of the witch trial scene, Bedevere is seen releasing a dove to which he has tied a coconut. (Sure it's a dove and not a swallow, but Bedevere just isn't too bright.)
Next the Narrator makes reference to a swallow-flight as a measure of distance and compares it to the distance of a laden swallows flight (laden with a coconut, we presume). The cast gets a bit anxious and yells at him to "get on with it" before eventually shooting him with an arrow when he doesn't stop.
But the Pythons, immune to the archers of their own creation, still aren't done with swallows. In the Bridge of Death scene, the Bridgekeeper asks Arthur about the airspeed velocity of an un-laden swallow, a question that will lead to the Bridgekeeper being flung into the abyss. Moral of the story: don't ask swallow-related questions.
Why this swallow obsession? Well, everyone enjoys a running joke, the one you think is finished but manages to crop up again (and again, and again). But maybe the swallows also serve as an example of the trivial concerns we tend to get obsessed with at times, and tend to cloud our judgment. Should the guard really be so concerned about where Arthur got the coconuts, or should he focus on the fact that he's (oddly) using them to pretend he's riding a horse? Should the narrator explain distance in swallow-related terms, or should he "just get on with it"?
The swallow gag shows us that a) trivial things are seriously funny and b) sometimes we all miss the forest for the swallow-laden trees.