The biggest symbol in this story is probably the gigantic boar that's running around destroying everything. More than anything else, this boar seems to a symbol of wild nature, and the whole tale of the hunt can be seen to represent humanity's struggle to tame the wilderness around them. Did we just make up this theory for fun? Nah, we have our reasons.
For starters, the boar is one of Artemis's sacred animals. Artemis, being the goddess of the wilderness, is a symbol of nature herself, so any creature sent by her is probably going to have some of the same associations. The boar is also widely known as being a particularly tough creature to hunt; many a hunter has found the wrong end of a boar's tusk. (We're looking at you Robert Baratheon.) It seems to us that this animal's rough and rowdy reputation makes it a particularly good symbol of the untamed wilderness that surrounded the early Greeks.
We're guessing that tales of noble heroes conquering wild beasts (of which there are billion-jillion in every culture) were a major comfort to early civilizations. Tales like this were like pep talks for humankind; the subtext being something like, "Don't be afraid of the woods! Humans are awesome and we can conquer anything! Yayyyyy!". Of course, the Greeks were never that big on happy endings, and at the end of this story, the beast may be slain, but the hero Meleager is as well, showing that every battle has its cost and that nature isn't a force that can ever be fully conquered.