The reason why animals need continuous supply of oxygen is to survive. Almost all of the animals rely on the oxygen in order to be able to survive and function properly.
The oxygen is what enables the respiration, and the respiration is what allows for the cells to be alive and be able to function. Without constant oxygen supply, the cells will not be able to function, thus they will die, which will cause the whole organism to die.
Also, the oxygen is a gas that is used for the burning of glucose into fuel. This basically means that the oxygen is the final detail that enables the organism to obtain nutrients. If the oxygen is not present, than the organism will do everything to create glucose and it will manage to, but it will not be able to turn into a fuel for itself, thus it will not be usable.