An arms race, as with that between nations, is where a pair of rivals become locked in a spiral of offensive capability leading to increased defensive capability leading to increased offensive capability, with new layers of arms being developed and funded by a hugely wasteful expenditure on these weapons in order to remain in exactly the same place with respect to each other. Running fast to stand still.
Consider the cheetah and the gazelle in Africa (this is an example I've lifted from Richard Dawkins).
The cheetah has evolved speed and maneuverability to help it catch gazelles; the gazelle has evolved speed and maneuverability to help it avoid being caught by cheetahs. Both have an 'interest' in improving their speed and maneuverability at the expense of the other, and neither can afford NOT to improve their speed and maneuverability.
The the extent to which they can do this is limited only by the potential of their anatomy and physiology to evolve further in that direction without the radical reorganization which would involve temporary loss of some other function, giving a disadvantage which would be immediately selected against. So both prey and predator can evolve only in one direction, being driven by the evolution of the other and the selection pressure this produces in their respective environments.
Both cheetah and gazelle are literally running fast to stand still with respect to each other.
How could an intelligent designer arrive at this situation? Why would an intelligent designer create gazelles as food for cheetahs, then make it difficult for cheetahs to catch them? If gazelles had been designed as cheetah food, they would be slow and easy to catch, wouldn't they?
Similarly, if cheetah were designed for catching and culling gazelles, why did they need to overcome the ability of gazelles to avoid being culled?
Neither species gains anything from the extra effort involved and the investment they make in developing the ability to out-maneuver the other, other than in the context of the other's presence. No intelligent designer would create a system whereby one of its designs tries to make it difficult for another of its designs to work properly, and so needing otherwise pointless and wastefully competitive modification.
About the only reason an intelligent designer would have for designing cheetahs and gazelles and then making them try to out-evolve one another in a wasteful arms race, would be if it enjoyed watching blood sports. But then, as an omniscient designer, wouldn't it know the outcome of the chase in advance anyway?
Another example from Africa is the Acacia tree and the giraffe. Trees are naturally selected for taller trunks by the shorter ones being eaten by giraffes. Giraffes are selected for longer legs and necks to avoid starvation by increasingly tall acacia trees.
What does the acacia tree get from having to grow a very much longer trunk than it would otherwise need if there were no giraffes? What does the giraffe gain from having to grow longer legs and a longer neck than it would otherwise need if acacias were shorter? The entire purpose of an acacia tree is to produce acacia trees and it does this by producing flowers and seeds. This would work just as well at ground level if only there was nothing around which regarded the seeds as food.
So why would an intelligent designer make it difficult for the acacia?
And why would an intelligent designer design giraffes to eat acacias, then place them almost out of reach?
Would any intelligent designer of a motor car design it so the gasoline it uses could avoid being burned so the car had to develop more and more elaborate and expensive ways to get the gasoline from the tank and the gasoline had to develop more and more elaborate and expensive ways to avoid being consumed?
Would this be intelligent design? Would YOU buy a car designed by such a designer?
Clearly, as these arms races demonstrate, there is no intelligence involved at all. What we see is exactly what we would expect if the process were an unintelligent, utilitarian process, selecting only for those solutions which work best to produce more of the next generation and having no ultimate goal in mind and with no concern that the process might be leading to the eventual extinction of one or both protagonists.
What we see is exactly what we would expect to see if there were no intelligent designer but only an evolutionary system driven by natural selection.