Back a LONG time ago, Hydaelyn (the mothercrystal) and Zodiark coexisted. But Zodiark got too big for his britches and Hydaelyn smacked him down (sealing him in the moon). But in doing so, she "shattered" into 13 pieces. These pieces became alternate dimension versions of our world (the Source). Those alternate dimension versions of Hydaelyn also served as a "barrier" to keep Zodiark sealed away.

However, on the 13th shard, a flood of darkness came (caused by an overly aggressive Ascian) and engulfed the world, turning it into "the Void", a world devoid of aether and all it's inhabitants becoming twisted monsters.

The First is another one of the shards (their all named for a number 1-13). That is where we are going for Shadowbringers, because it was hit with a flood of light, which is just as bad as the flood of darkness. Luckily, the flood was halted and we have a chance to fix it.

The overall story, however, is that the Ascians (masked evil guys) are causing the Calamities (major destruction) on the Source (our world) because every time they do, Hydaelyn absorbs one of the shards to heal herself. They have done so 7 times so far (we have had 7 Calamities). If they can cause 1 more Calamity, that will be enough to weaken the barrier and allow Zodiark to free himself.