Hmmm...
...you know...essentially, it makes sense, even. Hit Points represent the amount of hits (with a certain damage rating) that are needed to bring your character down. They also represent the ability to soak up damage, endure pain and keep going. This ability goes up with levels, as the experiences of fights and taking hits strengthens your character to do those things.
A 1st level fighter, with not much fighting experience and not much tolerance for wounds and pain, takes a hit from an axe. He takes it more or less in one big wound, and it gives him a great amount of pain, enough to nearly disable him combined with the wound itself. His friend the cleric casts "Cure Light Wounds" on him, closing the wound nearly completely, and assuaging the pain it caused.
A 5th level fighter is able to better "distribute" the hits all over his body through experience instead of taking it on one central point. Like that, the damage is less threatening and more widespread. It´s still the same amount of damage, and of pain, but our fighter is better able to minimize the threat to his life and to stand the pain. After the fight, he´s healed by his friend again, with a CLW. The spell still heals the same amount of damage as it did before, and stills the pain...but this time, the amount of damage it heals wasn´t as threatening to the fighter´s life anymore, and the pain he was more used to.
How´s that sound?