Does a flight briefer require a license in the US?
I imagine that the do receive plenty of training and I belive there was a time where they were "Flight Service Specialists" (this may have changed when LM took over). The FAA offers a great free course through their WINGS program on Flight Briefings and how they work on the other side (from the briefers stand point). I attended it recently and I recall that the woman giving the course was a former briefer and was a Flight Watch Specialist. I don't recall if that was a requirement or not thought (you can always reach out to her and ask). It should be noted that Flight Watch/Lockheed Martin Flight Services are now run by Lockheed Martin on contract to the government and are no longer run solely by the government as they once were, which may have changed things operationally (I have only ever known it in its current state).

As for staying on top of things, in todays modern times that is more the job of a computer than the briefer. Things like NOTAMS and AIRMETS/SIGMETS in route are given to them digitally once you tell them your course/route. There was once a time when this would have been a manual process but those days have long since passed. In my general experience flying VFR on a longer flight the briefer will not give you all the radio outage NOTAMS up front but will ask you if you "want to hear the radio nav aid notams" or something like that. They will give you things with in a certain distance from your planned route or if you ask for it. For example you can say "is VOR XXX in service?" and they will look it up for you. 

Other Notes:


There is an interesting discussion on the day to day here. 
You can read up on the history here.
And this Article