This is called 五回のなぜ in Japanese ("Go-kai no naze," or Why Five Times) and refers to root cause analysis. Of course the number of times the process need be repeated until the root cause is identified is variable. Five in this case is merely a jingly mnemonic, as Japanese people are so fond of.
I learned this working in a manufacturing environment for several years the was implementing the Toyota Production System and all the things that are a part of it like Kaizen, Kanban, etc.
After I left I thought it was too bad that all that good process was locked into the manufacturing world. I've seen it creep in more and more, often under the classification of UX with understanding user flows and actual problems of usability on a site (another area where asking why 5 times is very helpful to get to the root of a user's feedback).
I've also seen what I call "kanban flavored agile" be one of the most effective engineering processes for environments with ever changing requirements who don't need the rigid deadlines of sprint-based planning. When I saw it in manufacturing it was all about "just in time delivery" meaning a factory could easily switch production lines when suppliers missed deliveries (or delivered defective goods) or a machine broke down, etc. Which when you compare to the sorts of things that happen most every day in a typical software organization doesn't sound all that different.
I've always thought the idea behind "5" was to encourage you to ask more than once or twice. I think it's natural to stop after once or twice but you usually get the good stuff further along. Why is the sky blue? The atmosphere. Oh, ok -- I got it. There's a lot more to the story.
In Japanese, you often encounter words/phrases that sound more or less the same but are written differently.
If we're just going for homophones it could just as easily be "fifth floor" but that also doesn't make a lot of sense as a pun.