Question
why does the clippings in the garbage can (provided by our city for green waste) stink so bad>?

Answer
All dead organic matter decompose. This applies to plant parts, animals, humans etc. The decomposition process is carried out by a number of bacterias of which some require oxygen to be present to work. When grass clippings are piled, oxygen is almost eliminated due to the "compact" and "wet" nature of clippings. Since the clippings are also very high in nitrogen (compared to straw, or fallen tree leaves in fall for example) you usually end up with a bad smell. The nitrogen rich products require extra oxygen and when not available, an anaerobic (means "without oxygen") decomposition process takes place by another set of bacterias. This process is not as good and fast as the oxygen rich decomposition process, but it will accomplish the same result in the end. The result is a bad smell, however.

I recommend that you do not rake the grass clippings off the lawn (do not bag either). Let the grass clippings fall down on the lawn and decompose where they lay. Using a mulching mower is much recommended. It is very beneficial to the lawn.