Organic produce tends to quickly spoil as it is generally not treated with preservatives or chemicals, nor is the produce usually covered with wax. Organic produce also ripens very quickly and either has to be consumed or frozen before it goes bad. Keeping up with ripening schedules in a vegetarian household is not always an easy task, especially when you run your own business or keep the kind of hours we do.
The theory behind the bags is simply that they absorb ethylene gas from the produce. Plants use ethylene as a hormone and one of its actions is to ripen fruit. If you can reduce the concentration of ethylene around a banana, for example, it shouldn't get over-ripe and mushy as quickly.
Ripening is not necessarily what causes spoilage in produce. Berries usually develop mold long before the fruit itself goes bad and may even spoil before becoming ripe.
Do Green Bags work? I have found that, in our household, they do. Again, however, we do buy organic, start with good, fresh food, and thoroughly wash the produce before it goes into the bag. We also let things ripen outside of the bag first, although we have, while traveling, put bananas into a bag while they are still a bit green.