It's pretty easy. Every turn you accumulate some amount of culture which gets you to milestones that will cause great people to appear. For example, if you have no wonders or buildings, you'll probably get 5 culture per turn for your palace (you get 0 in Democracy, so Greeks need some cultural improvement or another government to take advantage of this bonus).
More great people just adds 50% to your accumulation. So if you're bringing in 10 culture per turn, you'll add 15 every turn. It doesn't make your people more cultured, but means the GP will show up more frequently.