How does a neutron star collision turn mass into gravitational waves? I heard on the TED Radio hour that in the neutron star collision that LIGO recently detected, some of the mass of the neutron stars was turned into energy in the form of gravitational waves. How does that work?
 A: This not too hard to understand. In fact, applying even Newtonian intuition will give an answer that is a reasonable (qualitative) approximation of the (always complicated) relativistic reality.
Simply put this can be viewed as two processes:


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*The mutual gravitational pull accelerates the two objects. In Newtonian terms, the gravitational forces do work as the inspiral shrinks (converting potential gravitational energy to kinetic energy).

*As the two neutron stars mull around in each other gravitational field, gravitational waves are created. (This bit clearly is relativistic and does not have a proper Newtonian counter part, but if you want you can think about how moving electric charges generate electromagnetic waves.) This process takes kinetic energy from the moving neutron stars and turns it into gravitational radiation.
This pretty much sums up what is going on. Of course, this is a bit of a simplification. In general relativity, things like kinetic and potential energy do not have nice, local definitions making the description much harder.
Note that unlike the suggestions made in the comments, the Baryon number of the neutron stars does not change during the inspiral. In particular, the energy emitted in the gravitational waves does not come from converting rest mass, it comes from increasing the binding energy of the system.
I realize that the two point above do not full answer the question, "How does a neutron star collision turn mass into gravitational waves?". The missing ingredient is that the gravitational mass of the system as a whole is given by
$$ \text{(total rest mass)} - \frac{\text{(binding energy)}}{c^2},$$
hence increasing the binding energy means reducing the gravitational mass.
A: The short answer is that no one knows. When you say mass is turned into energy in the form of gravitational waves, youre talking about mass in the form of nucleons (neutrons) being converted somehow into quanta of gravitational waves (gravitons). Viewed from this perspective, the question turns roughly into, "what is the quantum nature of gravity?", and that is an open question. The reason is that for us to understand the conversion of neutrons into gravitons presupposed, in some effective sense, a coupling between the two fields, in specific an operator in the lagrangian which generates a vertex containing the standard model fermions and some quantum gravitational field (normally thought to be some massless spin-2 field). But in reality such a theory is nowhere near being fleshed out entirely. No one knows if this is even the correct way to think about gravity in a quantum sense. 
