As there are 3 different futures there are 3 different ways they came to be.
During the first version of the future (1.1-1.6) the future predators naturally evolved from bats over millions of years long after humans went extinct.
During the second future (2.1-3.10) the predators came to be after Oliver Leek used the predators as an army and one day they would escape and kill everybody. When Oliver Leek was killed, the same future still occurred. Helen Cutter learned that Cutter would at some point create the future predator in the ARC and they would escape and kill everybody. Helen killed Nick to stop this but the same future still occurred. Helen learned that Christine Johnson was to blame as she was stealing predators and cloning them. Therefore she was now the one responsible for cloning predators and them escaping and wiping out the human race. The same future still occurred so Helen just decided to wipe out the human race.
After Phillip Burton took over 50% of the ARC, the third version of the future came to be where the earth became nearly lifeless and everything was dying. With humans dying off, future predators evolved to take advantage of the plentiful prey.