How does gravity escape a black hole?

There are some good answers here already but I hope this is a nice short summary:

Electromagnetic radiation cannot escape a black hole, because it travels at the speed of light. Similarly, gravitational radiation cannot escape a black hole either, because it too travels at the speed of light. If gravitational radiation could escape, you could theoretically use it to send a signal from the inside of the black hole to the outside, which is forbidden.

A black hole, however, can have an electric charge, which means there is an electric field around it. This is not a paradox because a static electric field is different from electromagnetic radiation. Similarly, a black hole has a mass, so it has a gravitational field around it. This is not a paradox either because a gravitational field is different from gravitational radiation.

You say the gravitational field carries information about the amount of mass (actually energy) inside, but that does not give a way for someone inside to send a signal to the outside, because to do so they would have to create or destroy energy, which is impossible. Thus there is no paradox.

Well, the information doesn't have to escape from inside the horizon, because it is not inside. The information is on the horizon.

One way to see that, is from the fact that nothing ever crosses the horizon from the perspective of an observer outside the horizon of a black hole. It asymptotically gets to the horizon in infinite time (as it is measured from the perspective of an observer at infinity).

Another way to see that, is the fact that you can get all the information you need from the boundary conditions on the horizon to describe the space-time outside, but that is something more technical.

Finally, since classical GR is a geometrical theory and not a quantum field theory*, gravitons is not the appropriate way to describe it.

*To clarify this point, GR can admit a description in the framework of gauge theories like the theory of electromagnetism. But even though electromagnetism can admit a second quantization (and be described as a QFT), GR can't.

Let's get something out of the way: let's agree not to bring gravitons into this answer. The rationale is simple: when you talk about gravitons you imply a whole lot of things about quantum phenomena, none of which is really necessary to answer your main question. In any case, gravitons propagate with the very same speed as photons: the speed of light, $c$. This way we can focus simply in Classical GR, ie, the Differential Geometry of Spacetime: this is more than enough to address your question.

In this setting, GR is a theory that says how much curvature a space "suffers" given a certain amount of mass (or energy, cf Stress-Energy Tensor).

A Black Hole is a region of spacetime that has such an intense curvature that it "pinches out" a certain region of spacetime.

In this sense, it's not too bad to understand what's going on: if you can measure the curvature of spacetime, you can definitely tell whether or not you're moving towards a region of increasing curvature (ie, towards a block hole).

This is exactly what's done: one measures the curvature of spacetime and that's enough: at some point, the curvature is so intense that the light-cones are "flipped". At that exact point, you define the Event Horizon, ie, that region of spacetime where causality is affected by the curvature of spacetime.

This is how you make a map of spacetime and can chart black holes. Given that curvature is proportional to gravitational attraction, this sequence of ideas completely addresses your doubt: you don't have anything coming out of the black hole, nor anything like that. All you need is to chart the curvature of spacetime, measuring what happens to your light-cone structure. Then, you find your Event Horizon and, thus, your black hole. This way you got all the information you need, without having anything coming out of the black hole.