Core activities, like many things in RPG, are about motivation. What do players want, and how do they get it? A good core activity helps a GM harness what players want to create situations that are interesting in play.
Dungeons & Dragons has a very good core activity. Players, like a lot of people, like power. They like being able to do cool things and have control over their world, or in this case, the game world. D&D has power in spades, in discrete, easy to use packages -- spells, class abilities, magic items, and in later editions feats -- and a direct, simple path to it. Collect XP, level up.
It gives the PCs a reason to go out and do things. Whether its rumors of a dungeon filled with peril or a glimpse of some guy getting mugged in an alleyway, XP gives extra bite to any plot hook. It also makes it easier for PCs with very different motivations to get along. The guy who's out for revenge against his father's killer and the guy who mostly just wants to get rich are both willing to go along with the do-gooder's latest crusade, because there'll at least be a level in it for them, and maybe a neat magic item. Whether leveling up is an end in itself or something that will help them achieve other goals, XP gives everyone something in common.
Now, each edition has a slightly different take on how you get XP and exactly what those powers do, and I'm going to work out some of those details in later posts, but that's the idea at the heart of them all. It's a major part of why the game has kept going for so long. And thanks to computer games, it's an idea that will likely outlive D&D itself. Just another debt we owe to 1974.