Lets put it simply: Predators eat other animals for food.
Predators prey on other animals at all stages in their lives, from those who feed on eggs in their nests to the great hunters who track fully grown animals to the opportunistic who prey on the old or injured and the scavengers who feast on the remnants of the recently dead. The role of the predator in evolution is to drive it forwards by removing evolutions dead ends or by providing niches in which scavengers thrive.
A predator which is better suited to its role as a predator is more likely to survive long enough to breed and have offspring. A predator which is not well suited to its role as a predator is less likely to survive long enough to breed and have offspring. Through this mixture of natural selection and random chance predators have evolved over thousands of years. Random variations in their genetic code cause minor changes to their physical form or behavior which are then judged by natural selection and random chance, those considered successful will produce offspring and some of those offspring will have those same adaptations, repeat this over many generations and you have the evolution of the predator.
The same thing happens to prey species, they are judged by natural selection and blind luck but they have the disadvantage that the predators described previously are trying to eat them. If a prey animal is born with a genetic variance that makes it slightly different than other members of its species, just like with human baby's every animal born or hatched is slightly different, this will either make it more vulnerable to predators or make it less vulnerable to predators. That variation will either make it easier for the creature to survive or harder for it to survive and those which survive long enough will have the opportunity to produce offspring which will have the opportunity to carry that adaptation forwards. As with anything in life their is always a certain amount of luck involved!
I should point out that their are few, if any, predator species which are not themselves prey to another species, one of the few exceptions being humans but even we occasionally fall prey to another member of our own species or another creature.
I was walking up my street one morning to work and i spotted a bird on a roof, i nearly missed it because it's feathers matched the color of the roof tiles so perfectly. Imagine that was a random variation, a one off mutation, which made it harder for that individual bird to be seen by predators when it was sat on house roofs and made it harder for its own prey to spot it. This would give that bird a distinct advantage over other birds of the same species and because of this it survives and produces six offspring of which three have the same color feathers. Of the three birds with normal color feathers only one survives long enough to produce offspring but two of the roof colored birds survive long enough to produce offspring. If on average each bird has six off spring and half of the roof colored birds offspring have roof colored feathers then the 3rd generation of roof colored birds will be six strong. Over a number of decades a species can adapt, or evolve, to suit a niche environment, in this case the urban environment, and maximize its chances of surviving in that environment.
That was a modern day example of the predator's role in evolution, simply put predators eat evolutions dead ends!