Sometime it good to revisit the basics.
The question asked over on the RPGSite was "Why do Fantasy Games use Levels".
The short answer is "Because of D&D". That led naturally to the question where to do levels in D&D come from?
The answer is found in Chainmail. When the fantasy supplement was added Gygax needed a way to represent the heroes found in myth and novels. The route he choose was to represent them as X number of fighting men. So a Hero was worth 4 fighting men, and a Super Hero was worth 8. Wizards are not as neatly pegged but also have distinct levels of ability from the Seer to a full Wizard. The genesis from Chainmail to D&D saw this simplistic setup expanded by Arneson and Gygax into the full range of levels we see in the original edition of D&D.
Levels have endured over several editions of D&D and been imitated in other RPGs as well. The reason is that it is a easily grasped concept by the novice gamers. With the confusion of combat, character generation, spells, monsters, etc Level stands out as a clear and easy to use method of comparing the power of two character and denoting advancement. That simplicity allowed it to endure despite it early uses not being very realistic or amicable to character customizations