How and when did this practice originate? This article from The Atlantic presents some of the basics:
In fact, the former Halliburton subsidiary of Kellogg, Brown and Root (KBR) consummated its veritable marriage with the U.S. military during the Clinton administration, when the firm’s logistical capabilities were indispensable to the Balkan interventions that many liberals supported. The KBR-designed military bases in Bosnia and Kosovo became templates for those in Iraq and Afghanistan.
Rather than mercenaries who will fight for the highest bidder, private contractors like KBR and Blackwater are composed mainly of retired American noncommissioned officers (NCOs), working alongside the same military to which they used to belong. Just as other professions tap the wisdom and expertise of retirees, so does the American military. Indeed, some contractors, like Triple Canopy, are known to hire veterans of the most elite Special Operations units in the U.S. military. “I’m hiring the elder statesmen of the combat arms community,” one Army colonel told me, referring to some private contractors he was taking on to supplant his uniformed troops in a noncombat capacity. “They won’t have to go through any sniff test when they arrive in the field as consultants. They’ll be instantly looked up to.”
The use of military contractors can make a lot of sense from the perspective of economics: it all depends on the difference between fixed and variable costs. Professional soldiers represent a fixed cost to a government (and, by extension, its taxpayers): they will get paid the same salary regardless of whether or not they are called into combat (ignoring, for the sake of discussion, extra costs such as combat pay). Contractors represent a variable cost: they only get paid if they get called into service. Thus, in a world in which a government will not know in advance how much military service it will require, it can minimize its expenses by holding fixed costs equal to the lowest level of constant expected demand, and using variable costs to handle peaks of demand. The actual minimum depends on the price differential between the cost of professional soldiers and the cost of hired contractors: expensive contractors, relative to professional soldiers, means it can be cheaper in the long run to hire more professional soldiers and reduce the use of contractors. The minimum also depends on actual levels of combat: unexpected increases in combat levels can force the use of costly contractors.