About the $sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect)

We know that $sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect) can be calculated by the Berry curvature, but we also know that $sigma_{xy}$ is calculated by the properties of the body!

Why it can be explained as edge state to some extent? I know edge-boundary correspondence, but what is the physics in it? I hope that is a elegant explanation about that.