From a mathematical point of view, quantum mechanics can be formulated in the language of a non-commutative, unital C*-algebra $\mathcal A$ (of observables).
In this context, what does the C*-property
$$\vert\vert A^*A \vert\vert = \vert\vert A \vert\vert^2 \qquad \forall A \in \mathcal A$$
mean from a physical point of view, i.e. which physical oberservation is modeled by requiring this property?
I think this property cannot by given a physical interpretation in a nice way, as usually for measurements only selfadjoint operators are concerned, as pointed out by @Norbert.
However, I can give one mathematical explanation why this is important: The $^*$ is an involution on the banach algebra, and often, we like think of it as taking the adjoint operator, but in general it isn't, especially when the banach algebra is not an algebra of operators in a Hilbert space.
For instance, it's not clear that $\Vert A \Vert = \Vert A^*\Vert$, but the property above can be used to show that:
$$\Vert A \Vert^2 = \Vert A^* A\Vert \leq \Vert A^* \Vert \Vert A \Vert, $$ thus $\Vert A \Vert \leq \Vert A^*\Vert$ and the opposite direction follows with $ {A ^*} ^* = A.$
Thus, one could summarize in an overly dramatic way: The equation makes sure that the involution $^*$ behaves like one would expect from taking the adjoint.