The intro takes place in the past, well the war does. The reason for the Garlean empire fighting the Dragons is unknown.
The rest of this post is pure speculation and theories:
My guess is, the empire wants something that the dragons possess (maybe the power to summon the eidolons/summons/espers/etc that are linked to the aerethyte stream (underground force just like lifestream in FFVII)).
Further I speculate that the four people with tattoos on their necks, that we meet in the opening sequences for each nation. (male white-haired huyr (ul'dah), miquote(limsa), lalafell and female huyr(gridania) are the four warriors of light destined to protect the planet and the aerethyte stream from the garlean empire who wants to abuse its power. (similar stories have occurred in multiple FF games). This is all just speculations created from playing all FF games and general knowledge of Square-enix story telling,. We will probably find out that the empire is evil, the dragons are an independent force trying to protect the planet and we ( the adventurers) are thrown in the middle to observe the epic battle unfold.
FFXI 2003-2007: Elaron ~Phoenix
FFXIV 2010-> : Gus Morgan ~Wutai