There are some things about old-school D&D that I've completely changed my opinion about at some point, and the system of limiting demihuman level advancement is one of them. For many years I thought that it was completely stupid, an arbitrary game-balance artefact that gave demihumans some weird kind of mental disability that stopped them from learning anything new after a certain point. In fact, until recently I'd intended to house-rule level limits out when I ran Labyrinth Lord.
What changed my mind was the sudden realisation of what level limits mean. They're not a “learning difficulty” - they're a performance plateau, like when an athlete reaches a peak performance where they just can't get any better, and actually have to work hard just to keep up what they have.
The "peak performance" claim might seem strange when the level limits look far short of the level that the game goes up to (20), but look at things in perspective. Compare the demihumans in the LL core to Fighter, the human class that they all have much in common with. All but one of them can get beyond Name level, by which a Fighter already has most of the hit points he's ever going to get, and near-superhuman combat ability overall. At level 9, without any other modifiers, a Fighter can wound an opponent in plate armour with shield 50% of the time; he can confront large numbers of level 1 Fighters (each one a “Veteran” per the D&D level titles) successfully, fight and kill creatures like giants and dragons of colossal scale, and is so renowned for his prowess that merely founding a small keep will attract warriors from far and wide to pledge themselves to his service. In short, a level 9 Fighter is a warrior of epic proportions. A Dwarf at level cap (12) is beyond even this, with slightly more hit points than the equivalent level Fighter and significantly better saving throws. The Elf can also get beyond Name Level (to 10), and is fully as capable at magic as a Magic-User of equal level as well. Even the humble Halfling gets almost to Name level (8).
Imagine that these breakpoints represent the maximum physical potential of the demi-humans. Dwarves are excellent at combat thanks to their toughness and determination, but they they're just not built for much speed or agility. Elves have speed and agility to spare, but lack brawn. Halflings can get surprisingly effective in combat, but are disadvantaged in physical power. In all cases, building on their strengths eventually reaches a point of diminishing returns where it doesn't overcome their inherent weaknesses anymore. Hence, the level limit. Humans are the special case – they are the supremely versatile race, whose unique package of qualities allows them to develop in potential without limit. Outside of the context of RPG game systems, the idea that someone who already is exceptional among the elite (at any physical or mental discipline) keeps getting better and better is extraordinary; so really the demihumans aren't odd for having level limits, it's the humans who are odd for not having them!
Looking at things in this way is especially in-genre: it's a very common trope in fantasy and sci-fi that humans have the greatest potential because of their all-round capabilities, unhindered by any intrinsic limitations. It also works well on a world-building level – having the very mightiest warriors, spellcasters, and so on be humans helps to justify the human-dominated world that D&D (and most classic fantasy) is built around.
Primarily, though, the level limit mechanic is about game balance. It gives demihumans a disadvantage to compensate for their racial benefits. Many other games model “human versatility” by making them equally suited to any class/role while non-humans excel only in one or two roles, failing to consider that in group games, characters typically do best by focusing in one or two roles anyway because they're in a team of specialists. The level-limit method handles “human versatility” on a micro-scale (versatility within a role) not just a macro-scale (versatility across roles) which works much better within typical RPG gameplay.