The Hit Point Mechanic is just a way - and abstract indeed - of setting a challenging threshold for when the hero buckles. If you want something more lethal or something that inflicts penalties as the hero's situation worsens, feel free to houserule that or use a different system.
But ask yourself: What makes a hero a hero? That he has a great destiny? Well, then consider the Hit Points a "destiny buffer"... Is it that he's toughened up and has learned how to survive various dangers? Then call them "survival points".. Do the Hit Points represent real wounds and blood? Then find a way to explain how the hero can contain 20 gallons of blood.
Likewise, healing will be reflected by this explanation of Hit Points. Pick one that you like and roll with it. The idea that hit points is a combination of luck, survival instincts, endurance, and perhaps even divine favor, destiny, or force of will is a simple and easy-to-explain way of making the abstraction of Hit Points plausible. And mind, it's a way that makes Hit Points work without too much fuss so you can focus on more important elements in your game, such as the actual story, for instance.