At its heart, the Persephone's story is meant to explain the turning of the seasons. When the goddess of spring, Persephone, makes her yearly descent into the world of the dead, her mother, Demeter, spreads winter over the earth. When Persephone returns to the world above, spring returns with her. This tale was a way for the ancient Greeks to grapple with a phenomenon that must've been pretty darn mystifying. Hey, want to see a video on why we really have seasons? Check it out here. If only the ancient Greeks had YouTube.
Pretty much every human being ever born has spent some time worrying about how death. We ponder how it's going to happen, what the larger meaning of it is, and where (if anywhere) we're going to after we finally kick the bucket. It's really no wonder that, in so many of the ancient myths, somebody ends up going down to the world of the dead and somehow ends up coming back again to the land of the living. People just love a resurrection story. Maybe it gives us hope that we may live on (in one form or another) after we die, or maybe it's a reminder that, even if we pass on, the cycle of life will continue long after we're gone.
In the story of Persephone, Demeter, and Hades, Persephone ends up symbolically dying when she goes to live with Hades in the underworld, then she's symbolically resurrected when she returns to her mother in the land of the living. The whole world mirrors her death and resurrection, because when she goes to Hades, winter arrives and the plants die. When she returns to the living, she brings spring with her.