I have to admit that power-laws (in general) used to be my shtick so I am happy to shed some light on their general importance in physics which obviously also hold for astronomy.
The main idea of a power law is nicely written in Wikipedia, but the essential part is the I highlighted in the following quote:
[A Powerlaw is] a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities
The (mathematically) nice part is exactly this, that over a really large range of the x-values, the y-values follow the same dependency. Usually, "really large range" means values spreading over 3 to even 10 powers of ten.
When fitting power laws in practice, there are effects on the sides of the scale, meaning for small and large $x$-values, where usually attributed to "finite size effects".
Further reading
- There is the research flavor Self-organized criticality (SOC) which "is considered to be one of the mechanisms by which complexity arises in nature". One of the founding fathers (and mothers) was the late Per Bak. His book How Nature Works is accessible for laypersons and IMHO a good read.