So for a specific parameter space, the behavior of these superconducting circuits can be simplified to a harmonic oscillator (which operates as a two-level system). Simplifying a complex system to a well-known, smaller system is crucial in many other fields of physics. Can you think of another example?
Harmonic oscillators are useful to model diverse systems that oscillate around a point of stable equilibrium. The main idea behind this is to get rid of higher order terms in the physics that describes certain phenomena, to only analyse the terms that matter the most. Hopefully these terms will be enough to describe what you are studying.
Think of describing the movement of the earth around the sun. If you don't need many details, analysing the forces between the earth and the sun from a classical perspective can give you a lot of information already. This way you would conclude the orbit of the earth around the sun is circular, which used to be a pretty good answer back a couple centuries ago.
If you decide to include the moon, then you get a 3 body problem that is not easy to solve, but it helps you describe other phenomena and get more precision in your results.
Of course you could also use general relativity instead of classical mechanics, and then you would be able to describe even more fascinating physics, like black holes.
Another example you may be familiar with are lattice vibrations in solid state systems. Here classical normal modes are used to describe phonons. However, if you wanted to study particular effects, you would need a more complex model to reproduce them.