Massive amounts of Xana (magic) are stored within Xana Crystals. These crystals are orb-shaped, and many of them are mistaken for relics or jewelry or the like. Only one was truly known to the public: the Xana Crystal being studied at Nuva Academy on Nuva Island.
The Gyth know that there are eight Xana Crystals, and they need all eight for their special plan. The first one allowed some of them to permanently enter our realm (the realm of Radia etc.). With each successive Crystal activated in their posession on Zeris Island, they can bring more Gyth into our world and inch ever closer to their ultimate (or one of their ultimate) goal(s).
When a Xana Crystal is stolen by the Gyth, the entire region suffers a consequence linked to the Goddess or God whom the Xana Crystal represents. The Nuva Crystal represented some sort of ocean deity, hence the island sunk. The Roth Crystal represents the Goddess of Fertility, so local pregnant women began having misscarriages and other women could no longer get pregnant. The Georoma Crystal will cause the volcano to erupt (per the God of Fire).
The first Xana Crystal they collected was the one on Nuva Island. They had a 4-hour window to obtain it. WIth it in their posession when they returned to the Gyth realm, they were able to return to Radia (our realm) a few months later (albeit in small numbers).
They seized the second Crystal at Roth Temple. More Gyth will arrive in a week or so, but they want to grab the third crystal so they send an army of drones (and crack Gyth) after the third crystal at Geo Island.