Growing degree-days are a measure of accumulated heat during the growing season. They are calculated by taking the daily maximum and minimum temperatures, determining an average, and comparing that to a base temperature. Growing degree-days directly relate to the growth and development of plants and insects. Below certain temperatures, no growth occurs (or it happens extremely slowly). At higher temperatures, development proceeds rapidly (unless it's so hot that stress sets in).
What this knowledge brings to your garden and landscape is a better predictor of when your plants will be in a certain stage of development or when insects may attack. Weather patterns can vary from year to year, so a calendar date prediction is not reliable for temperature-dependent events. For example, your forsythia shrub may bloom 2 weeks earlier this year than last if it is warmer. By the same token, the aphids that attack it likely will show up 2 weeks earlier as well.