In the 1600s, England had a nasty civil war, between the Royalists (Cavaliers) and the Parliamentarians (Roundheads), in which both had their own armies. The Royalists had operated under the past rules, in which armies were established by the feudal responsibilities of vassals to their lieges - that is, the King would draw up an army by calling in his responsibilities to his lords, who would then conscript their vassals, and so on and so forth. This would have been a "Royal Army".
On the other hand, Parliament established an army by going around and conscripting soldiers into their own individual armies, eventually coming up with Cromwell's New Model Army. This army proceeded to kick the crap out of the old feudal army, and take off Charles I's head in the process.
When the Restoration came about, all of the good generals that were still around were veterans of the New Model Army, and they didn't hesitate to let the new king know that. This was also around the point where Parliament began to pass the Mutiny Acts, which essentially gave it control of the standing army, since under the previous law, there had been no law governing the existence of an army during peacetime.
Essentially, you had a situation where the army was "owned" by the people through Parliament, while still nominally under the Monarch. That's what this interpretation is saying the Second Amendment is trying to do - have the President ultimately be Commander-in-Chief, but the States in control of the Army.