What is the physics behind “Bulk-edge correspondence”?

There is a frequently mentioned concept in the field of topological insulator called "bulk-edge correspondence" or "bulk-boundary correspondence", which basically gives the relationship between the total number of edge states and the topological properties of all the bulk bands below the gap. Can anyone tell me what is the physics behind this "correspondence"?

In other symmetry classes and dimensions, it is not always clear (to me) what the quantized quantity refers to, but at least for $\mathbb{Z}_2$ time-reversal-invariant two-dimensional topological insulators, it refers to a kind of "time-reversal polarization", and for chiral insulators for chiral polarization, but these interpretations are vague.