The Thundercats didn't go to Earth.
Third earth is Earth in the Future.
This is the sort of thing a child will miss even when it's basically stated, but an adult will see long before it's directly said. ThunderCats takes place on a planet called Third Earth. To a kid, that sounds like it can't be our Earth. We live on the first one, the third one must be out in space somewhere. However, if you pay attention, the show quickly makes clear that "Third" is a matter of time, not space. This is Earth in its Third Age, when humans are no longer the dominant species and in fact many of the planet's inhabitants are alien colonists and refugees from across the cosmos. This fact is made especially clear in the episode "Excalibur," when Merlin makes a guest appearance and reveals that the King Arthur stories took place on the same planet where the Thundercats now live.
https://www.looper.com/103557/things-thundercats-notice-adult/?utm_campaign=clip