I agree that it doesn't sound too surprising. Intuitively, if we have a system with local interactions, it's reasonable to suppose that the only relevant interactions between two subsystems only occurs at the boundaries, and therefore the mutual information scales with the area (this is also the intuition that the paper uses in Figure 1). However, maybe I can motivate the importance of the result:
- The specific numerical factor in the inequality is notable. Not only is the mutual information of two subsystems bounded by some value proportional to the surface area, but it is the same value for a wide variety of cases.
- By verifying this bound rigorously, one can also better understand the necessary assumptions that go into it. A notable quote from the paper is
So, indeed, we get an area law for the mutual information solely from the existence of a length scale $\xi_M$, which expresses the common sense explanation of Fig. 1. This area law is also valid for zero temperature and when violated immediately implies an infinite correlation length $\xi_M$.
In other words, the area law can be violated in certain cases, and now we better understand why that is.