Each species reduced the carrying capacity of the environment for the other. The predator eats the prey. So, the predator numbers increase because the predators are being well fed. But, the numbers of prey decease because they are being eaten. Soon, there isn't enough food for the predators so they begin to die.
Fewer predators means that the prey can now increase in number.
This process continues as a cyclic increase and decrease in predator and prey numbers. A classic example is that of the fox which eats rabbits. Trappers would bring in pelts and the Hudson Bay Company would buy them and resell them to furriers. The Hudson Bay Company records do show a remarkable, long lived cyclic behavior. But, in the period between 1875 and 1905, the cyclic change goes in the wrong direction.