The Player's Handbook (1989) for Advanced Dungeons and Dragons, 2nd Edition says
The human race has one special ability:… Humans can choose to be of any class… and can rise to great level in any class. The other races have fewer choices of character classes and usually are limited in the level they can attain. These restrictions reflect the natural tendencies of the races (dwarves like war and fighting and dislike magic, etc.).…
The limits also exist for play balance.… The demihuman races have other powers that make them entertaining to play—including the ability to be multi-classed…. (20)
The Dungeon Master's Guide (1993) for Advanced Dungeons and Dragons, 2nd Edition says
[H]umans are more motivated, indeed, driven by ambition and the desire for power, than the demihuman races. Thus, they advance further and more quickly than others.…
Demihuman characters are limited in how high a level they can achieve both to preserve internal consistency (humans are more flexible than non-humans) and to enforce game balance. (14)
So the ability to multi-class is, in essence, paid for mechanically through demihuman class and level limits and justified narratively by saying that humans possess greater ambition and drive than demihumans. This can be lumped together: human drive limits humans to dual-classing rather than multi-classing because humans throw their all into things.
(Going into this, I'd thought at the time that the humans' short lifespans were part of the the reason that they couldn't multi-class, but that detail's absent from the Dungeon Master's Guide and goes unmentioned in the entries for shorter-lived races in the Complete Book of Humanoids.)