How did the greeks explain the four seasons?

Demeter, the goddess of harvest and plants, had a daughter called Core or Persephone.

One day Persephone was out collecting flowers when Hades, god of the underworld and the dead, saw her and fell in love. He went to Zeus, king of the gods, and demanded that he be given Persephone as his wife but Zeus didn't want to give her to him because Persephone was not only his niece but his daughter as well so he told Hades that he would think about it. Hades went back to the underworld but the next day he saw Persephone again and decided that he couldn't wait any more and took her to the underworld to live with him. Demeter was very unhappy to lose her daughter so all the plants on Earth began to die, and she went to Zeus to demand her daughter back from Hades. All the people on Earth were starting to die from the lack of food so the gods called an emergency council. Zeus told Hades that he had to give Persephone up but one of Hades's gardeners told Hades that he had seen Persephone eat 7 pomegranate seeds and the law was that if you ate the food of the dead you had to say in in the land of the dead for ever. But the gods realized that the world would die if Persephone didn't return so Zeus decided that Persephone had to stay in the underworld one month for every seed she had eaten.

We get summer when Persephone is home, Autumn when Demeter is sad because her daughter has to go soon, Winter when she is in the underworld and Spring when Demeter is looking forward to her return.