All the floating landmasses are cuased by Rocks with high concentrations of Aetherically charged Wind crystals. This actually has an effect on what can grow on them to since its commented in the Botany quests that many plants struggle in soil with so little Aetheric earth and such large concentrations of Aetheric Air.

The reasons for it happening are probably rather varied. In some cases it likely caused by disasters, in some cases its almost certainly artificially caused, and in some cases it seems likely to have happened naturally. Nym for example was caused by a disaster of overaspected wind crystal. My personal theory was this was a desperate attempt by the Scholars of Nym to save the city from the flood of the 6th Calamity in the same way the Mhachi used the Void Ark to escape the flood. We know that there is a scholar credited as saving Nym but we aren't told how.

The Isles in the Sea of Clouds seem to be more likely to be natural. The Moggles at least have been living up in the sky for quite a long time. Their legend of King Moogle Mog mentions that some Moogles were sent to find a home in the lands below after 'Fire rained from the Skys' and destroyed the landmass they lived on breaking it up into pieces. Wether this is just refering to the Churning Mists and the Fires were a reference to the start of the Dragonsong War or an older story refering to an earlier calamity isn't clear but there were Moggles living in the Sea of Clouds at one point.