E. Gary Gygax notes in several places that the class limits and level limits were both game balance and to force the game to be humanocentric.
I'll let EGG speak for himself (Dragon #29, Sept. 1979, p. 12):
The character races in the AD&D system were selected with care. They give variety of approach, but any player selecting a non-human (part- or demi-human) character does not have any real advantage. True, some of those racial types give short-term advantages to the players who choose them, but in the long run, these same characters are at an equal disadvantage when compared to human characters with the same number of experience points. This was, in fact, designed into the game. The variety of approach makes role selection more interesting. Players must weigh advantages and disadvantages carefully before opting for character race, human or otherwise. It is in vogue in some campaigns to remove restrictions on demi-humans — or at least relax them somewhat. While this might make the DM popular for a time with those participants with dwarven fighters of high level, or eleven wizards of vast power, it will eventually consign the campaign as a whole to one in which the only races will be non-human. Dwarves, elves, et al will have all the advantages and no real disadvantages, so the majority of players will select those races, and humankind will disappear from the realm of player character types. This bears upon the various hybrid racial types, as well.
While I disagree with his conclusions, His point is pretty clear: He felt it important to nerf demi-humans so as to force players to choose humans in preference to them.
It's been asserted that Gygax ran high-level games, citing Tomb of Horrors. The S1 module with vermillion cover lists no levels on it for players, but includes mostly done pregens; the green cover shows levels 10-14. The highest level characters included are a 14th level cleric and 14th level mage; the lowest, a halfling fighter/thief 4/5. Definitely more evidence that Gary's idea of High level was rather low by modern 36-leve (BXCMI), 30-level (4E) or 40 level (3E/3.5E) games.
Frank Mentzer (ExTSR) makes an important point in understanding why the initial restrictions were so severe... From his comment in response to this answer:
Remember that when the RPG biz began, roleplaying a non-human was a major stretch. Up until then, in assorted wargames & minis games, everybody was playing humans, period. We've broadened a lot over the years.
It should be noted that Chainmail was one of the earliest minis games to have non-human hominids, and strongly appears to be the earliest, with a 1971 release. (Source: Board Game Geek database multiple searches on 25 May 2011.) D&D follows in it's Chainmail roots.
To put in perspective, the original little book set only covers to about level 15. Gary Gygax commented that, generally, his campaigns only ran to about levels 12-14. Old Grognard on RPGNet, who played in Gary's group, confirms this as well, as did Dave Arneson. The concepts of level in many newer editions are very different now than what Gary was using in his own games. Just looking at AD&D & Moldvay/Cook vs earlier editions shows level creep.
Likewise, several other early game designers generally considered 9th level time to hang up the armor and settle down. T&T only covers to level 15. The Arcanum to level 20. Palladium to level 15. Bushido to level 6.
To show the progression of the level limits even during EGG's day; multi-class combos allowed shown as slashes. C=Cleric F=Fighter, Mu=Mage, T=Thief P=Paladin Mk=Monk A=Assassin D=Druid. The Number is max level, a hyphen indicates maximum with high stats, and parenthesis indicates NPC only.
OD&D, no Supps (V1 p6-8):
Human: F∞ C∞ Mu∞
Dwarf: C7 F6-8 T∞ F6/T∞
Elf: F4 Mu6 F4/Mu6
Halfling: F4
OD&D + Supp 1:
Human: F∞ C∞ M∞ T∞ P∞
Dwarf: C7 F6-8 T∞ F6/T∞ P6-8
Elf: F4-6 Mu6-9 C6 T∞ F4-6/Mu6-9/T∞ P6-8
Halfling: F∞ T∞ P6-8
Half-Elf: F6-8 Mu6-8 T∞ F6-8/Mu6-8 F6-8/Mu6-8/C4 P6-8
Note: Due to wording on page 8, no racial restriction exists on Paladins are present; that was errata corrected promptly...
Also note that the number of demi-human options grew exponentially
OD&D + Supp 1 & 2:
Human: F∞ C∞ Mu∞ T∞ P∞ Mk16 A13
Dwarf: C7 F6-8 T∞ F6/T∞ P6-8
Elf: F4-6 Mu6-9 C6 T∞ F4-6/Mu6-9/T∞ P6-8
Halfling: F∞ T∞ P6-8
Half-Elf: F6-8 Mu6-8 T∞ F6-8/Mu6-8 F6-8/Mu6-8/C4 P6-8
Note: No errata for S1 was included...
OD&D + Supp 1, 2 & 3:
Human: F∞ C∞ Mu∞ T∞ P∞ Mk16 A13 D12
Dwarf: C7 F6-8 T∞ F6/T∞ P6-8
Elf: F4-6 Mu6-9 C6 T∞ F4-6/Mu6-9/T∞ P6-8
Halfling: F∞ T∞ P6-8
Half-Elf: F6-8 Mu6-8 T∞ F6-8/Mu6-8 F6-8/Mu6-8/C4 P6-8
Note: No errata for S1 was included, here, either
AD&D 1E
all multiclasses are limited in level to the same levels in each class as single classed.
Human: C∞ D12 F∞ P∞ R∞ MU∞ T∞ A13 Mk16
Dwarf: (C8) F7-9 T∞ A9 F/T
Elf: (C7) F5-7 MU9-11 T∞ A19 F/Mu F/T F/MU/T MU/T
Gnome: (C7) F5-6 Il5-7 T∞ A8 F/Il F/T Il/T
Half-Elf: C5 F6-8 R6-8 MU6-8 T∞ A11 C/F C/F/MU C/R C/MU F/MU F/T F/MU/T MU/T
Halfling: (D6) F4-6 T∞ F/T
Half-Orc: C4 F10 T6-8 A∞ C/F C/T F/T F/A
As an aside, exceeding AD&D racial caps was allowed starting with the Unearthed Arcana sourcebook.