The inmates of this dimension are cast into the dimension and reside in a featureless state of existence from which they observe the regular dimension clearly. They do not age, not need any substinance and largely not affected by the regular dimension.
As such, they survived the destruction of Krypton and focused their attention on Earth seeing as how most of the surviving Kryptonians now reside there. Most have a particular grudge against Superman seeing as how his father created the method of their own damnation. As such, they usually cause destructive trouble when they manage to escape while Superman struggles to return them to the Zone.
The only exception is Mon-El, a Daxamite youth who met Superman when he was Superboy on Earth and discovered he was acutely vulnerable to lead poisoning. To keep him alive, Superboy cast Mon-El, with his permission, into the Phantom Zone where he resided until the 30th century where Braniac 5 of The Legion of Super-Heroes developed a cure to allowed him to leave safely.