I must admit that when I first started looking at the rules for older versions of D&D, I found the idea of race as class strange and laughable. I get this same reaction from a lot of gamers when I broach the subject. One of my friends here gets a bemused look on his face whenever he mentions “that version of D&D where Elf is a class.”
Look, I'm not drinking the Kool-Aid on any one version of D&D being “superior.” One is certainly free to prefer whichever version of D&D one likes best, for whatever reason. Please don't pretend that the things you like about your favorite flavor of the game makes it objectively better than other versions. That's a tired, tired philosophy and I have no interest in discussing it. Your opinion is perfectly valid, but that doesn't mean you should dismiss a game mechanic out of hand. Speaking of opinions, here's why I've come around to liking race-as-class in D&D.
Since the first edition of AD&D and right on up through 3E D&D, I think a character's race has mattered less and less in favor of his character class. When I sit down at the gaming table, it's easy for me to forget that so-and-so's character is an elf or halfling or whatever, all I can remember is whether he's a ranger or a cleric. It's to the point that race almost completely doesn't matter and I think that's a shame. Look at Tolkien's stories, certainly the inspiration for D&D's demihuman races as we know them. If I asked you, “What is Legolas?” your first answer is probably “He's an elf.” Elf, you see, is a viable archetype, despite what more modern RPGs would have you believe.
It's important to remember that D&D is an archetype-driven game. Especially in earlier versions of the rules, but certainly present in all editions to one level or another, a character's class is his most important trait. In the absence of a codified skill system, how do you know whether your character can perform some task that's not defined by the game rules? Character class, that's how! Even with a skill system in place, a lot of what a character can and can't do is hard-coded into his class. Magic-users possess the sorts of abilities that magic-users should have acquired after a youth and early adulthood spent learning to cast spells.
In the first version of D&D there were 3 classes: Fighting-Men, Clerics, and Magic-Users. In addition, a player could choose to be an elf (which could switch between Fighting-Man and Magic-user each adventure), a halfling (which could be fighters), and dwarves (fighters). While at first glance you might think that there is no difference between a Fighting-Man and a dwarf, you'd be mistaken. The dwarf also gets a rather substantial bonus to saving throws against spells; access to a powerful type of magic weapon; the ability to note slanting passages, traps, and new construction underground; and the ability to speak Gnome, Kobold, and Goblin languages.
To the modern gamer, this might not sound like much to give the dwarf. Bear in mind, though, that this original version of the game didn't grant any benefits for high ability scores beyond an XP bonus. That means that your Fighting-Man with the 18 STR didn't hit any better than a similar character with a STR of 11. (Except that the former character got an XP bonus, meaning he'd level faster than the latter character, thus making him a better combatant over time.) Above all, OD&D was very light on rules and open to player improvisation and DM fiat. That means that any bonus ability, no matter how small, could have a big impact on a character's power.
Now take a look at Labyrinth Lord, the retroclone game patterned after 1981's Basic version of D&D. Here we see the races codified almost as classes of their own. At the core, the dwarf is still just a fighter, with the same hit die and combat abilities. However, the dwarf has slightly different XP requirements to gain each level, a different set of saving throws, and some extra abilities. For all intents and purposes, it's a class all to itself. While the game says “dwarf” is your class, you could just as easily say you're a “dwarven fighter” and the only difference would be semantics
The other common argument against race as class is based on roleplaying. Of course not every dwarf in the world is exactly the same "class". NPCs don't need to follow the same rules as Player Characters. Indeed, they probably shouldn't most of the time. Just because all dwarf adventurers fall into the same class, that doesn't mean there aren't dwarf cobblers and cartwrights and candlestick makers. It also doesn't mean that the DM can't, should he so desire, invent more class options for dwarf players to choose from. Think that your world should allow dwarven clerics? Great, make up a class specific to your milieu and write it out.
When AD&D rolled around, Gary Gygax introduced the idea of racial modifier to ability scores. This is the concept that elves are more agile than humans and thus receive a +1 bonus to their Dexterity (and a -1 penalty to their Constitution due to a more fragile build). While this sound okay on paper, in practice it seldom amounted to much. AD&D's ability scores determined a bunch of secondary abilities for a character, such as his Strength affecting his bonus to hit with a melee weapon or his base chance of bending metal bars. However, there were big areas within each ability score where a +1 bonus made little mechanical difference in the character's stats. In most cases, the guy with the 12 STR wasn't mechanically stronger than the guy with the 10 STR. Unless your stats happened to fall at a specific number, a +1 either way didn't make a big difference. In other words, elves didn't seem any more dextrous than humans.
Third Edition remedied this issue somewhat by changing the ability score system so that every 2 points a stat increased, it gave a bigger bonus. Now race mattered a little bit more, but it was easily forgotten once character creation ended. Plus, unless you rolled really well (i.e. a legit 18), only luck kept the humans from being just as dextrous as you.