Extinction happens when a species loses the required population numbers to reproduce. A recent example is the Eastern Mountain Lion which was declared extinct last year because there are no monitored mating pairs left in the wild. Another case is the California Condor which should be let go. Also the Ivory Billed Woodpecker in Gulf Coast areas was thought to be extinct due to logging until scientists spotted them quite recently.
So people bringing the animals back could refer to a number of things, like eliminating habitat reduction, hunting, etc like the case of the American Bison and Grey Wolves in Yellowstone. It also could refer to captive breeding programs like the Condor, which isn't going to work.