Killing other players will cause them to drop ducks. The higher the Duck Journal level, the more ducks drop when a player kills someone. Duck Journal Levels also cost $5 each in the Mann Co. Store (the Duck Token you already mentioned).
Journal owners can grab ducks to count towards your current Duck Streak (or start one if they don't have a duck streak going).
If you don't have a Duck Journal, they do nothing except make noises when you collect them. Usually, they make quacking noises, but the glittering ones will cause one of the Merasmus "Bonus Ducks" lines to play.
Duck Streaks are like a parody of Killstreaks. Like Killstreaks, messages will display on screen when a player passes certain numbers of ducks.
Duck Streaks reset upon death.
The duck journal is also an Action slot item. The higher your Duck Streak the louder it quacks when you hit the Action button.
This event goes away on January 5th... but since Valve added it as a Holiday to the game code, plugins (like the one I wrote) may be able to enable it again after it ends.