It doesn't. It escapes from outside the event horizon. The event horizon is the dark area in the middle.
At first glance, I think that the gravity information can't escape from inside the event horizon, just like light can't. That means that the event horizon describes a frozen version of the mass inside it, not a current "live" version.
And that seems to work, if you think about gravity waves. There aren't any changes to the gravitational field coming out from inside the event horizon. But it doesn't work so well if you think about gravitons. "There aren't any gravitons coming out" should be equivalent to "flat worldlines", which is very much not true just outside the event horizon.
It also doesn't seem to work for a situation like a black hole merger. The spacetime outside the event horizon is this frozen snapshot, but it can still do this spiral around this other black hole? That doesn't seem to make a ton of sense.
So I'm not sure my answer is very good. But it's a fascinating question. If anyone has a real answer, I'd love to hear it.
Remember that this applies everything where r≤1. As far as “flat” worldlines I think you might be thinking of a geodesic approaching r=1 in terms of a null geodesic, which isn’t necessarily true unless we’re dealing with a photon or graviton. Regions I,II of the Classic Kruskal-Szekeres extension illustrates this pretty clearly.
https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates...
The total mass of the black hole (and all other information possible about it) can be described in terms of the boundary at r=1, so there’s no problem with mergers or accretion. To answer Wallace’s original question, we see no information escaping the black hole. What we’re seeing is sort of like shining a light on an absence of information, and observing the shadow cast. That’s not quite right, but it’s close.
This discussion might help where my ability to answer your excellent question is failing: https://physics.stackexchange.com/questions/937/how-does-gra...
The same problem would exist for the electric field from a charged black hole. However, static fields don't need propagating photons to establish them, so you don't have to get photons from inside the black hole in order for the electric field to be established outside.
The same would be true of gravitons. But one respondent indicated that general relativity can't do a second quantization like electromagnetism, and therefore gravitons are... suspect? Impossible? Not proven? It wasn't clear to me how strongly to take that statement.
I noticed my karma is jumping down in waves. Unrelated comments are all being downvoted at the same time, suggesting unlikely coincidence or that some users are clicking on my profile and going through downvoting all the comments.