That comes straight from pg, as reddit was his idea. He sold it to Alexis and Steve as "the front page of the Internet", that is, a user-curated list of all of the most interest links out there. While it may be read as ambitious, it was originally just intended to describe what reddit was and how it worked. It was the precursor of the "AirBnB for lightbulbs" taglines that dominated pitches a couple years ago.
Alexis and Steve applied for the first YC batch with a cell phone food idea that pg rejected.
He told them that he liked them, but not the idea. He suggested instead that they build "the front page of the Internet". They did, and the rest is history.
People have a tendency to think that phrase is somehow bragging, but it's really just what the original idea for reddit was. It's not about their size, but about the intended purpose of the site.
Reddit is living up to it now, though.