The title refers to a wine made with dandelion petals and other ingredients, such as citrus fruit. (I found numerous recipes for dandelion wine on the internet.) In the story the dandelion wine made by Douglas's grandfather serves as a metaphor for capturing the fleeting joys of summer. The bottles of homemade dandelion wine brewed during the summer allow the season to be savored and remembered for a longer period of time.
"Hold summer in your hand, pour summer in a glass, a tiny glass of course, the smallest tingling sip for children; change the season in your veins by raising glass to lip and tilting summer in."
~Dandelion Wine, Ray Bradbury