How did the sweatdrop originate?
There doesn't seem to be any "official" announcement for that, but I think it's safe to say that since Manga and Anime make heavy use of symbols to re-create a concept or a situation, the sweatdrop functions the same way. In other words sweatdrop (see image below) is a reference to the fact that when people feel uneasy or feel similar emotions, they usually get sweaty. You don't get that sweaty in real life, but again, they're symbols.  

More symbols that might prove this point:


The cruciform popping veins are a clear reference to the veins that pop out and can be seen pulsating when you're nervous or you're about to lose it.


 Source


The red cheeks when you're embarrassed (blood flowing), the red nose when you're drunk (yes, nose has that tendency sometimes when you get drunk).
The saliva drop when you're hungry. Your mouth does get watery in that situation, because it anticipates the food you're about to eat. Mastication is the first stage of digestion.
The eyes shaped like >.&lt; that indicate nervousness or excitement. (I bet everybody can picture this face expression).


And so on. There is more (as a start) on the Wikipedia page for Manga Iconography.