There is no proof of bulk-boundary correspondence for topological phases in general. In fact, topological phases like toric code model does not have gapless excitations on the boundary.
For non-interacting fermion systems protected by internal symmetries (as in the "periodic table" classification), bulk-boundary correspondence holds. For non-interacting fermion systems protected by a spatial symmetry, gapless surface states also exist on those crystal surfaces preserving the symmetry.
In some sense, the existence of some kind of boundary states is all there is about topology in non-interacting fermions.
The TKNN (bulk) and Büttiker (edge) explanations for the quantized Hall conductance correspond to different geometries.
In the TKNN theory, the "sample" consists of a torus closed on itself and therefore has no edges at all. In this case the electric potential is uniform, and the electric field is due to the time derivative of the vector potential (it lasts only as long as one varies the magnetic flux inside the torus). In this case, the Hall current is truly a bulk current.
Büttiker, on the other hand, considers a Hall bar with different electrochemical potentials on each side. If one (as does Büttiker) assumes that the electrostatic potential is uniform within the central region of the bar, and rises on the sides, then one finds that the current flows along the edges (in opposite directions), with more current flowing along one edge than along the other because of the different chemical potentials.
In a more realistic description, the electrostatic potential is not uniform in the bulk of the bar, so that the current flow takes place both along the edges and within the bulk of the bar. In any case, the net current is completely independent of the actual profile of the electrostatic potential accross the hall bar, and depends only upon the chemical potential difference. That is why why Büttuker and TKNN obtain the same answer for the (quantized) total current.
A nice discussion of this question is given by Yoshioka.
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I take your question to mean "topological insulators" in the larger sense of all symmetry classes and all dimensions (which includes the quantum Hall effect case).
In this case, referring only to the quantum Hall case, the edge quantity and the bulk quantity physically mean the same thing: electrical conductivity in direction perpendicular to applied electric field. Thus, the physics is that this conductivity is the same. For systems with an edge, one interprets the current to be running along the edge, whereas for systems with no edge (bulk), one interprets the current to be running within the sample. In real-world samples, which of course have an edge, there is a certain amount of both types of currents (the ratio depends on the boundary conditions and "working function"), and anyway the quantized conductivity of either mechanism indeed agree.
The way to make the connection between the two quantized numbers is via the fact that we are assuming that in the part of space where both systems exist, their Hamiltonians are the same, so that the edge Hamiltonian may be considered a "restriction" of the bulk Hamiltonian up to some (spatially bounded) arbitrary boundary conditions.
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.