It is an abbreviation of "Hōkoku" (報告, to report), "Renraku" (連絡, to inform) and "Sōdan" (相談, to consult), and is more memorable as a homonym of hōrensō, the Japanese word for "spinach".
The origin of "Ho-Ren-So" comes from Tomiji Yamazaki who was the ex-president of Yamatana Security firm.
In 1982 he started using the term for effective business communication in his firm and it has been widely spread and used throughout Japan through his book titled Strengthen Your Company with Ho-Ren-So.
Additionally, if information is made known to all members of a team, ownership of the schedule and tasks can be divided up.
Finally, if a subordinate proactively asks for information, a decrease of quality will be prevented and likely won't occur again.