Why is it that black holes emit Hawking radiation? Black holes are basically neutron stars with such a gravitational force that even light cannot escape from it. But what causes it to emit Hawking radiation?
 A: 
Black holes are basically neutron stars with such a gravitational force that even light cannot escape from it. 

A black hole is a mathematical solution. A neutron star over the critical mass gets so dense that it forms larger and larger time dilation relative to the outside universe thus we get to see what happens on short time scales.

But what causes it to emit radiation

Note that classical general relativity does not predict Hawking radiation. So the answer has to be a quantum effect. So recall that we see things that happen in short time scales?
So things would have to emit radiation in short time scales and if they normally interact with other things in a short time scale it must balance out but the asymmetry and the geometry of the set up makes it not balance out.
Another completely different way to look at it is through the equivalence principle and conclude that an accelerating particle detector clicks more often than one moving at a steady velocity, the Unruh effect. And this just means that what looks like a vacuum in one frame does not look like a vacuum to another frame accelerating relative to the first.
A: The reason black holes emit radiation is because virtual particles are popping into existence and popping out of existence throughout space including at the event horizon of a black hole.  When they pop into existence they pop into existence in pairs that then annihilate within a fraction of a fraction of a second.  When a pair of virtual particles pop into existence near a black hole one may fall into the black hole while the other escapes and is unable to annihilate with its twin and so it becomes a real particle that carries away energy that was originally in the black hole.
A: 
Why is it that black holes emit Hawking radiation?

We don't actually know that they do. Hawking radiation remains a hypothesis. It's been around for so long that people rather take it for granted, but there's no actual evidence for Hawking radiation. And when you read the "given" explanation, it doesn't seem to make sense. See Wikipedia: 
"This radiation does not come directly from the black hole itself, but rather is a result of virtual particles being 'boosted' by the black hole's gravitation into becoming real particles".
But see anna's answer here: virtual particles exist only in the mathematics of the model. They are abstract things, not something that can be boosted into reality. That's a fairy tale. So is this:
"An alternative view of the process is that vacuum fluctuations cause a particle-antiparticle pair to appear close to the event horizon of a black hole. One of the pair falls into the black hole while the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole)." 
Electrons and positrons do not pop into existence, and neither has negative energy. We know of no negative energy-particles. That's another fairy tale. One that totally ignores gravitational time dilation.   

Black holes are basically neutron stars with such a gravitational force that even light cannot escape from it. But what causes it to emit Hawking radiation?

They aren't neutron stars, but nevermind. We have good evidence that black holes exist, for example there's something very small and very massive in the centre of our galaxy. And general relativity is one of the best-tested theories we've got. And general relativity features a variation in the "coordinate" speed of light. The light doesn't escape a black hole because the coordinate speed of light at the event horizon is zero. Hawking radiation totally ignores this. 
