>>64123397 (OP)
It's about the mental load on an officer and the number of "things" he can track in real time. After centuries of experimenting, the general consensus is that 3-5 is the optimal range of "playing pieces" a CO can juggle without dropping one. Down at the lowest level, a fireteam leader manages 3-4 men. A squad leader manages 2-3 fireteams. A platoon commander has 3-5 squads, totaling about 30 men, which also fits a natural break point of being about the maximum number of men you can corral by running around and screaming at them. A company has 3-5 platoons for roughly 100 men, which is about how many you can physically keep an eye on in detail on the post industrial battlefield and scream at someone to scream at the guy you want screamed at. Then battalions, brigades and/or regiments, divisions, and finally armies all follow through on the same principle.
It's about breaking down the insanely complicated task of moving tens of thousands of men around in an environment where people are trying to kill them into discrete chunks, so that every decision-maker in the chain has a workload that fits neatly into his mental capacity. Each step up the chain sees a larger picture, but at a lower resolution.