Sorry, I missed that. The time direction does indeed become spacelike inside the black hole. The most remarkable consequence of this is that the singularity is at the future of every test particle that crosses the horizon.
This can be seen best in a conformal diagram or Penrose diagram (now that he has a Nobel prize might as well use the name of his creator).
You should track the r=0 line, initially it points upwards (it's timelike) as it chugs along at the center of the star as it collapses gravitationally. When the BH forms, it becomes horizontal (spacelike) and lies at the future of every test particle that enters into the BH. The singularity is inevitable for anything that crosses the horizon.
The Black Hole will at some point have radiated all its energy in Hawking radiation at which point it disappears and r=0 becomes timelike again.
Unfortunately it's hard to make it easier to understand without indulging in some math.
> In what sense does [time] become spacelike? Can one move back and forth in time inside the event horizon?