Why would anyone care about the moment of inertia of a planet?
Knowing the moment of inertia can teach you about the composition, density, and spin rate of a planet.
Here are a few reasons to find the moment of inertia of a planet.
You want to know what is inside: Since the moment of inertia depends on both the mass of the planet and the distribution of that mass, knowing the moment of inertia can tell you things about the layers of a planet, their density, and their composition.
You want to know how round it is: Round things have a different moment of inertia than oblong things or potato-shaped things. This can be useful for figuring out things like what the planet is made of, how much mass it has and how fast it is rotating.
You care about its rotation rate: The moment of inertia of the earth changes due to things like glaciers melting and earthquakes. The changes the rotation rate of the Earth slightly, making a day slightly shorter or longer. Thus, we have to add leap seconds to keep time measured with atomic clocks synced up to the day from the Earth's rotation.