From what I understand of it the singularity is analogous to what happens in the 2D Cartesian plane with functions such as f(x) = 1/x. When x equals 0, the function itself "breaks down" because the y-coordinate extends to infinity (ie f(0) is "undefined"). In the context of a black hole that means that neither time nor space (hence neither does matter) "exist" at this singularity. In other words upon arrival the falling object has essentially "reached the end of time".
