My understanding (which very well could be mixed with my headcanon, fair warning) is that a calamity triggers such an immense disturbance in the aether that it breaks down part of the dimensional fracture that Hydaelyn inflicted on the source. When that happens the anchor that kept the shard separate is lost so it "rejoins" with the source and becomes one with it again.

So each calamity is gradually undoing what Hydaelyn did originally. Once 13 calamities happen the source will be back to how it originally was and the seal on Zodiark will be broken.