Culture.

The Europe/Asia split originally referred to the sides of the Aegean in Greek geography and got expanded on from there. (The Roman province of Asia was just parts of western Anatolia; eventually it became "Asia Minor" and "Asia Major" was the larger group.) As for why the Caucasus and Urals, I'm sure it's because they're the most convenient borders in the area, but I don't know when they were selected.

Oceania is kind of a new one: there was an Australia long before either Antarctica or the present island were discovered because of the idea the Earth needed a counterweight at the antipodes. Even after everything was discovered, people usually called Australia the continent and everything about it was just grouped with it or Asia. (As far as I know, the only part of Indonesia that is connected to Oceania is on New Guinea and the line is [edit: sometimes (see comments below)] more cultural than geographic, as shown by the fact that the Indonesian half of the island often isn't considered part of Oceania.)

As for why the Greek version became standard, it's because the Age of Exploration started from Europe and not Mexico or Korea. The Arabs also cribbed the Greeks. Traditional models in China and the rest of the Sinosphere usually had a single land mass (divided into 9 main areas) prior to getting more accurate maps from the Jesuits, who simply translated the Latin names into Chinese.

In Russia and Japan, though, there aren't seven continents: they group Eurasia as a standard thing. Similarly, in Latin America, I think it's standard to group the Americas, which is part of why they get so annoyed at using "American" to refer only to los Norteamericanos. I'd agree that it's weird to group the Americas while keeping Europe and Asia distinct, but again it's cultural and historic.

You may be interested in some of these previous questions:

• For how long have we had the concept of "Continents"? When did Europeans started calling themselves "European"? How were the borders of what we know as "Asia" and "Europe" set up?

• Why are Europe and Asia considered separate continents, and was there ever a time when they weren't?

• When were the continents first given names, what were those names and who gave them those names?

• When and how did the idea of the 7 continents come about? Specifically: why is Europe a separate "continent" while the Middle East, a similarly culturally distinct area, is not?

• When and from who did each of the 7 continents get the name they have today? Have there ever been disagreements about the borders of the continents?

And now I'm off to add a new section to the Popular Questions pages!

EDIT: Done! "The number and names of the continents"