"There are viable theories and there are natural and elegant theories. However, all viable, natural and elegant theories contain dark-matter axions" - Ann Nelson. The axion is a hypothetical elementary particle. Without the dark-matter axion, physics has two mysteries to explain: The hidden mechanism suppressing the expected large amount of CP violation in the strong interactions (the "Strong CP Problem") and the nature of dark matter. Originally postulated to exist in order to solve the CP problem, the axion also happens to be an extremely attractive dark matter candidate. The axion is the puzzle piece that allows these two mysteries to fit naturally into our universe without using scissors and a permanent marker.
The axion was originally postuated to exist as part of the solution to the "Strong CP Problem". This problem came about from the observation that the strong force holding nuclei together and the weak force making nuclei decay differ in the amount of CP violation in their interactions. One solution to this Strong CP Problem introduces a new particle called the axion. If the axion is very light, it interacts so weakly that it would be nearly impossible to detect. The ADMX experiment aims to detect the extraordinarily weakly-coupled particle.
There are some objects in the universe that we
can't see, but we know they are there from
their effects on other objects. For instance,
stars that
are very massive collapse when they die and become
black holes, with a gravitational pull so strong that not even
light can escape them. This means we can't actually see a black
hole. So how do we know a black hole is there?
One way is through their gravitational
interactions: since they exert a strong gravitational force, they attract nearby
visible stars whose resulting motions reveal
the black hole.
In a similar way, dark matter exerts a gravitational pull on normal matter. Observing the effects on normal matter is the reason we believe there is appreciable "dark matter" in the universe. For instance, from the rotation, structure formation, and shape of galaxies, we infer that the mass in a galaxy is dominated by dark matter, not normal matter such as stars and dust. In fact, it turns out that most of the matter in the universe has to be in the form of dark matter. There's just not enough luminous ( i.e. non-dark) matter to shape the universe to its current state.
The Axion comes in comes in as a possible answer to the question "what is dark matter made of?" The Axion is a neutral particle, so light wouldn't interact with it. The Axion, if it exists, would also be produced in nearly the right amount to explain how much dark matter that seems to be around today. The only way to tell if dark matter is made of Axions is to look for them.
There are two models of the Axion that ADMX is trying to either find or rule out. Those are the stronger-interacting KSVZ model and the weaker-interacting DFSZ model. They both agree that the Axion would have to be light and interact weakly with matter, which is why we have to work so hard to find one. ADMX is taking a slice of the "allowed" values for Axions and searching in that particular region.
The horizontal axis shows the mass of the Axion, whereas the vertical axis represents how
strongly the Axion would interact with other matter at that particular mass. The Axion does
have a range where we would expect to find it, meaning if it is not in that range then it
simply does not exist. We know that the mass of the Axion (and hence the coupling strength)
cannot be very large, or else we would have found them in other laboratory experiments and
also in astronomical observation of Supernovas. ADMX is designed to look into only a slice
of those proposed mass ranges.
The reason it's only a slice and not the whole range is simply due to the equipment. The frequency that is scanned by ADMX depends on the tuning rods and the resonant cavity. Making the apparatus able to scan a larger frequency range would have cost more and made the apparatus bigger, which makes cooling and transportation harder, among other things.
As to why it is that particular slice, it's because it's the most convenient one to look in. There's no significant reason to believe that the Axion would be more likely to be in any particular range, so this one was chosen based on it being easiest to scan with current technology.