> I'm not sure if it counts as a useful explanation to someone with a very general question

No, probably it isn't. However there are other readers. :-)

I'm sorry that my answer to the rest of your comment is going to seem very complicated and is certainly going to be very new to you. However, you did literally write "correct me if I'm wrong", and you are partly wrong. :-)

> mass->energy conversion

Special and General Relativity are very different when it comes to this sort of thinking, and unfortunately intuitions gained from experience with the former are usually counterproductive when actual gravitation is involved.

What we can do is draw a boundary around a region of spacetime and calculate the stress-energy within that region. That quantity will include contributions from matter, and from the matter's movement, and from quantities like pressure, stress and shear. The quantity is expressed in a sort of a 4x4 matrix called the Stress-Energy Tensor, and at every point in our region there will be a tensorial value.

In General Relativity, stress-energy is conserved; this is more general than the conservation of energy, and in many physically reasonable curved spacetimes the conservation is violated enormously on the largest scales, but stress-energy is conserved. In the case of our selected region of spacetime, it lets us say things like all the stress-energy that went into it in the past can exit it in the future.

That stress-energy includes the quantity that keeps the binary black holes from quickly falling onto each other. Typically you would want to work in a frame of reference in which that quantity is identified with angular momentum, but that's not obligatory in General Relativity.

We can instead do something like put ourselves above the barycentre of the binary black hole, rotating with the system, so that (wilfully blinding ourselves to distant stars and so forth) we do not see angular momentum, we only see something preventing the two masses from falling straight onto one another. This is making a choice of frame of reference in which we can work with fictitious forces to our advantage.

Einstein had an analogy here: consider two identical elastic spheroids alone in space, not too close to one another. One is exactly spherical, one is an oblate spheroid; they are arranged on a line that extends along the semimajor axis of the oblate spheroid through the centre of the exact sphere. An small observer at the midway point on this line between the objects can adjust its rotation so that either -- but not both -- objects appear to be rotating according to the observer. How does the observer decide if either object is the one the observer should be still with respect to?

Our observer can do perfectly reasonable physics if either or neither is rotating with respect to her. The choice often introduces fictitious (specifically, d'Alembert) forces into the picture; an example being centripetal or centrifugal forces.

The solution in modern physics is to be agnostic about whether either has "absolute rotation", and rely upon modern physics written in generally covariant form, and use tensors; or to bite the bullet and not be afraid of d'Alembert forces.

So we have just taken the latter path.

d'Alembert forces are encoded in the stress-energy tensor (an expert screaming "ARGH!" here should consider f^\lambda = -\Gamma^lambda_\mu\nu u^\mu P^\nu; whether you want to call them forces of affine connection, inertial forces, or whatever, they can be locally transformed away into a LFF frame).

So we have a fictitious-force quantity that's separating the binary black holes, and it's part of the local stress-energy, and we can export stress-energy from a region. We just need a mechanism.

Here we introduce another "ARGH!"-able thing: a stress-energy-momentum pseudotensor. We choose it carefully so that it can disappear in a local free falling frame, and we do so not so much for mathematical convenience but for conceptual utility.

For this comment, let's just call it a tool that lets us treat some of the left hand side of the Einstein Field Equations as the right hand side: we treat some of the "gravitational field" as if it were a component of the stress-energy tensor. Our goal is to transform our fictitious force separating the binary black holes into something that we can radiate out of the region of spacetime we've drawn around them. That something is identifiable with gravitational radiation.

So, the vanishing of the "force" separating the binary black holes as the black holes fall towards each other is tied to the emission of gravitational radiation. Moreover, as the binary black holes get ever closer, the amount of the force needed to keep them from just falling onto each other is ever higher, so the amount of gravitational radiation that is emitted at each tiny shortening of the distance between them goes up. When they are very very close together, the amount of gravitational radiation goes up to an extreme.

It is perfectly reasonable to say, "hang on, don't you have to keep adjusting the rotation rate of your frame of reference so that the black holes stay on, say, the X axis at all times?" Sure, but we're allowed to do that; in fact, it's almost encouraged in General Relativity.

> changing the shape of space

This you have right enough. Shifting contents of the stress-energy tensor outside our chosen region of spacetime must change the curvature of the spacetime in that region because in General Relativity spacetime curvature and matter must balance.

It's not quite that easy, though. Distances and lengths (which describe the geometry of spacetime) are encoded in the metric tensor. The arrangement of the stress-energy in our region determines the metric. Our region with the binary black hole sources some metric that we do not know exactly, but there are several approximate metrics that would describe the region reasonably well. Alternatively, we can take a well-known exact metric and perturb it based on the stress-energy in the region. We can use the post-Newtonian expansion in v/c. Or we can integrate the field equations numerically. Any of these approaches is reasonable, but one has to think about the two approaches a little differently, in order to avoid being misled about what is physical and what is not.