For many, many years we have seen women (or men) getting them placed on their eyes while receiving beauty treatments. Either in movies or in a spa, it is not an uncommon thing to see...especially in a good make-over montage. I looked into it as I have heard that it is BS from some and, truth be told, curiosity got the better of me as I'm one of those beauties that suffers from a little eye baggage.
As it turns out, there are benefits from using it and it is actually a quite simple explanation. They do, in fact, reduce puffiness BUT only due to the cooler temperature! That means they do work...BUT so do other cooled things like tea bags and cold compresses or those plastic gel-filled eye masks you throw in the fridge.
The only benefit that comes strictly from the cucumber is if you have a pre-existing eye irritation. There are antioxidants in cucumbers that do help relieve said irritation.
So, in short, sure they work but you could just as easily have a cold slab of meat on your face to get the same effect (which is typically also seen in movies after a fight to keep down the swelling on a shiner).
*I got information from the following sources: Huffington Post, POPSUGAR Beauty, the world's healthiest foods (whfoods.org)