Because they either have more abundant and or newer food sources to exploit or they have no or few natural predators to keep their numbers in check. When either of these happens, the introduced animal can easily out-compete native animals for resources.
With the European Rabbit. They easily easily thrive in Australia because of the lack of natural predators big enough to feed on them.
With the Red Fox, it's the opposite. They feed on animals that are not normally targeted by other animals. Such as crocodile eggs, small marsupials, nesting ground birds, etc. They have no other predators to compete with.