I'm a layperson who just read Brief Answers to Big Question and in the book, Hawking proposes that:
A typical black hole is a star so massive that it has collapsed in on itself. It’s so massive that not even light can escape its gravity, which is why it’s almost perfectly black. Its gravitational pull is so powerful, it warps and distorts not only light but also time. To see how, imagine a clock is being sucked into it. As the clock gets closer and closer to the black hole, it begins to get slower and slower. Time itself begins to slow down. Now imagine the clock as it enters the black hole—well, assuming of course that it could withstand the extreme gravitational forces—it would actually stop. It stops not because it is broken, but because inside the black hole time itself doesn’t exist. And that’s exactly what happened at the start of the universe.
I think I understand the notion of time dilation as the clock approaches the black hole but I'm having trouble wrapping my head around the idea that time literally stops inside the black hole. Hawking then uses this idea to propose that there was no time before the Big Bang:
As we travel back in time towards the moment of the Big Bang, the universe gets smaller and smaller and smaller, until it finally comes to a point where the whole universe is a space so small that it is in effect a single infinitesimally small, infinitesimally dense black hole. And just as with modern-day black holes, floating around in space, the laws of nature dictate something quite extraordinary. They tell us that here too time itself must come to a stop. You can’t get to a time before the Big Bang because there was no time before the Big Bang. We have finally found something that doesn’t have a cause, because there was no time for a cause to exist in. For me this means that there is no possibility of a creator, because there is no time for a creator to have existed in.
Does time stop at the center of a black hole? If not, why does Hawking present this idea? How else might can we describe the idea of "there was no time before the Big Bang"?