The notion you are referring to has to do with a particular set of coordinates you could use to describe the spacetime around a black hole. In that coordinate system the signature of the metric looks like (-+++) outside and(+-++) inside (roughly speaking, as it is not appropriate to think of these coordinates as t, x, y, z)1.

But the true insight of GR is that coordinates are not physical and that any physical question must have an answer that is coordinate invariant. So the whole exercise of imagining a "time direction" is moot. One can pose questions like "what do the null geodesics between two points look like as they move along their trajectories and one passes the event horizon" or questions like that. Ultimately only such questions have a physical meaning in GR and related theories.

Practically speaking in the local coordinates of an infalling observer passing the event horizon all the parts of the observer on the other side of the horizon, including signals, are simply destined to move towards the center of the black hole and the parts on the outside have some dynamical freedom not to. This is, in fact, what the inversion of the two coordinates means physically.

Footnote 1:

In Schwarzschild Coordinates the line element is given by: ds^2 = -(1 - 2GM/r) dt^2 + (1 / (1 - 2GM/r)) dr^2 + r^2 dθ^2 + r^2 sin^2(θ) dφ^2

Note that if 2GM/r is less than 2 then the sign in front of dt^2 is negative and the sign in front of dr^2 is positive. If 2GM/r is greater than one then the sign in front of dt^2 becomes positive and the sign in front of dr^2 is negative. This is the so-called inversion between a spatial dimension (r) and time. But it doesn't do to take this too seriously, as, again, the physics of the situation are entirely separate from any coordinatization we might choose.