When it says that "new" enchantments are x% stronger, what exactly does this mean? And does it stack additively/multiplicatively with the other enchanting perks?

Yes and yes.

Example: A enchant without any perks does 20 fire damage. You put all 5 perks in enchanter and then the 1 in fire enchanter for a total of 125% increase. You will end up enchanting for 45 fire damage.