My interpretation is a little bit different, and it relates to why Charisma is the key stat for turning/rebuking undead.
Turning/Rebuking Undead is a Divine Compulsion
If turning/rebuking undead were simply a matter of channelling divine energy, why wouldn't it key off of Wisdom, the main D&D stat for channelling divine magic? Turning/rebuking keys off of Charisma because the most important aspect to turning/rebuking is the character's force of personality. The divine energy and focus are necessary because it requires that divine energy for the character to do something they normally would never be able to do, which is magically compel undead (creatures normally immune to compulsion) to actionspecific actions.
This interpreation has the side effect of making the turn/rebuke undead mechanic more deity-agnostic, so to speak. A good god or a sun god would obviously devote a portion of its power make the ability to turn/destroy undead available to its clerics; just as a necromantic god would grant the ability to rebuke/control undead to their clerics. But what about all those other gods, neutral or otherwise, whose domains have nothing to do with the undead? Why do they specifically give their clerics the ability to turn/rebuke? The answer, in this interpretation, is that they don't. It is divine power, in the general sense, that allows clerics to do this through their sheer force of personality.
I would then describe the effects of turning/rebuking undead the same way I would describe any other magical compulsion effect. The good priestly character doesn't make the undead run away in fear, they command the undead to leave this area at once. And through the combination of the cleric's sheer force of personality and the source of their divine power, they obey.
Note this is by no means RAW (if it were the PHB/SRD would refer to the effect as a compulsion, and it also specifically states that it involves channelling positive/negative energy) but it is the way I have always interpreted it based on the reliance on Charisma.