Since ancient days, I have understood hit points to be something of a useful legacy of miniatures games in which each figure physically present on the game table represented more than one actual creature in play.
To wit, the example of a simple Norman infantryman, wearing a conical helm with nose-piece, a long coat of chain mail, and carrying a shield and spear. In the old days, a single metal figure of such would represent perhaps ten men. However, only two of these men would actually be considered to be on the line of battle at any given time. If the figure takes a hit, one of the men in the front rank is assumed to have been removed from the battle, and a man from the next rank is assumed to step up to take his place. Only after nine men have been removed from the battle, i.e. the figure has taken nine hits, will the combat effectiveness of the figure be impaired, and it will, of course, be eliminated entirely after it has taken ten hits. Figures representing more men required correspondingly more hits to be removed from battle.
This system was carried through, intact, when dealing with figures that represented a single individual. In such cases, the figure was assumed to be somehow tougher, stronger, and better able to "shrug off" damage than other men, but this was simply a gloss to explain why such a figure could, of itself, take multiple hits, where each hit was normally sufficient to drop a normal soldier.
The modern concept of hit points is a legacy of this system.
Rationalizing the effect of this in-game has never been an issue at our table: we take Gygax' word for it as quoted above. Warlord healing—most healing, in fact, including bard and cleric—is normally rationalized as the healer's encouragement allowing us to realize that our wounds weren't really as bad as we'd originally thought, e.g. it looks like a bloody mess, and had us worried for a bit, but the cut itself turned out not to be that deep.
—Siran Dunmorgan