As Joe mentioned, it is a product of combustion. The cold "moist" air is not necessarily drawn into the pipe upon shutdown. Air has the ability to hold more water at higher temps than at lower temps. So upon shut down, the pipes cool, and the exhaust gases cool, condensing water from the exhaust gases inside the pipes. Upon startup, you may get some of this water blowing out, but then more water is generated as the the hot "wet" combustion gases get cooled by the cool pipes and condense more liquid out. Once the exhaust system pipes are up to temp, less water will be condensed and it will be less noticeable at the exhaust. However, the air is still quite moist as can be felt by putting your hand or rag over a running exhaust.
Doug