The reason is that string theory is a proper S-matrix theory--- it defines things by probing at infinity using probes in the theory, and classical things using the massless fields at infinity as classical probes. The D-brane carries a mass density and a charge density, and when it is classical, the two are related as for an extremal black hole. This relation is also true in supergravity, where it is enforced by the partial supersymmetry of the state. You know the supersymmetry in both the supergravity limit and in the string theory, they are adjusted by both the number of branes in the stack and by the string coupling.
So it is natural to identify the two descriptions as weak-coupling and strong coupling versions of the same object. There are some small misstatements in the literature regarding the physical nature of the identification, it isn't local. So you see things in 90s literature like "the brane has a horizon surrounding it", or "the brane is behind a horizon". This is not a good statement, the brane description is valid exactly when the classical horizon picture is not, so you should identify the brane with the whole black hole up to the horizon, not with some singular sheet in the middle. The brane is what a quantum-scale black hole looks like when probed by long floppy strings. The brane description of a black hole is valid when the string coupling is weak.
The correspondence is motivated by the statement that any classical excitation in a gravitational theory is determined by its charges at infinity. The reason is physics--- if there were a separate "black brane object" and "D-brane object", you could throw a D-brane into an equally charged parallel black brane and get something different, but the degree of freedom count for RR-charged objects at small coupling is just the branes, so they are identified.
This physical insight is due to Polchinski, although it was not clear just how centrally important this insight was until Maldacena showed that the near-horizon correspondence between the classical near-horizon AdS geometry of the black hole and the quantum low-energy string-brane system maps string theory to gauge theory precisely.
The Chan-Paton factors come from the string actually making a junction with the black hole horizon. This junction has a classical limit--- it's a 1-d extremal black hole (the string) making an end point by having part of it fall into a higher-dimensional extremal black hole (the brane). This junction forming process is not really classical when it is described by Chan-Paton factors, that is at weak coupling, but it is classical in other limits.
Falling into an extremal black hole is classcially not possible without back-reaction--- a test particle can't fall into an extremal black hole because the horizon is infinitely far away. But a finite mass object can fall into an extremal black hole with no problem--- the backreaction makes the horizon come up to meet it (or if you like, it's already holographically projected on the horizon by the time it gets close, these are equivalent statements). A non-test particle object can cross an extremal horizon.
When you allow a long string to fall into a black hole, there can be a part that is external and a part internal. The Brane RR-charge and the string NS charge poke out to infinity in perpendicular directions, and they make it so that the endpoints have crazy classical monodromies, due to the electric magnetic duality. These monodromies are consistent with the interpretation that these intersections are described by the Chan-Paton factors in weak coupling limit.
There is a lot more here. The exciting different set of problems suggested by AdS/CFT took a lot of the wind out of the sails of these things, leaving many of the central black hole questions without a full resolution.