My take on the trailer and on the story is this:

First they show a couple of adventurers gathering guildleves (the main system of the game).
The main character (just like every one of us) is shown to have the Echo, hearing voices and experiencing a past event.
In this past event, a battle is show (i think it really has no relevance other than to show the game will feature common FF monsters - like Molboros - and to show battle regiments - with that fire arrow thing)

Then the background story begins. The character is then taken to the midst of a battle. Since those winged-cannon-thin-tall-freakish machines are being used by the garleans, I think that battle was an attempt of the Empire to destroy another city-state, just like Ala Mhigo. Maybe Ishgard. For me, it seemed as the dragons were protecting the land, and the empire then began killing all the dragons (and then began the meteor-shower scene, which were actually the dragons being shot down Q.Q).
However, they forgot about the king of all dragons (or mommy dragon XD) Bahamut, which appeared and destroyed the empire's dreadnaught. With Bahamut's sacrifice, the Echo was created and so were the primals. This happened 15 years ago. The official website also mentions the primals were first summoned around 10 years ago.

The adventurer's guilds were formed since the empire never got to Ishgard (or whatever city they were attacking). And everyone is preparing for the garlean invasion (maybe they are creating another dreadnaught D: 15 years seem like a decent time to create a thing with that size)

I'm really loving the story of FFXIV It has huge potencial. However, none of this is in the game as it should :s