The two words by themselves are common in the Japanese
language and when combined loosely mean "How and What
". Most commonly associated with Anime Kidou Senkan (Maritan Successor Nadesico)
which contained it's own little mini-series Naze-Nani Nadesico
which detailed certain aspects of the show or the equipment. Although it has been adopted in other places around the world as a way to explain something behind the scenes.
Sometimes more entertaining than the original show that they arte based on the can be done in omake style or with just plain silliness that doesn't occur in the show. Sometimes they will present different scenarios as to a branch in the plot. Nowadays, you can find references to naze nani in some other Anime or internet sites like Video Girl Ai or the webcomic Megatokyo to cite a couple of examples, sometimes in omake style.