What's the science behind making German potato dumplings (Knödel) fluffy but not fall apart?
Your dealing with a few things here. First is starches in general. The thing to know about starches is how they gelatinize and at what temperatures. 

This powerpoint is a nice primer on that topic. www.cfs.purdue.edu/class/f&amp;n630/gelatinization.ppt

Basically your dealing with amylose and amylopectin, together they are what we know as a starch. When they come in contact with water the starch cells begin to swell and when their gelatinization temperature is met they burst and release their contents into the medium they are in. In the case of a dough ball, your dealing with tons and tons of little cells being held together loosely at first by the physical pressure of kneading them into balls and then when the heat causes the starches to gelatinize they adhere to one another kind of like being caught in a net. 

Now the fluffiness portion of the question can really depend on how the dough is being cooked. For something like a dumpling being cooked in the boiling water, one would typically want to work the dough as little as possible to avoid making it too dense and if using flour to lessen the gluten formation that can make it very chewy like bread. Some recipes call for leaveners that can create gas bubbles when heated to a certain temperature and then through gelatinization the bubbles are trapped inside the dough and create an airy texture.