Here is the abstract of the original paper by Hawking:
In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature hκ2/πk≈10^−6(M⊙/M)∘K where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 10^15 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases where S is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.
Bold mine. It stresses that in classical General Relativity, nothing can escape the black hole horizon.
Why is the particle emitted viewed as the one with positive energy? What if, instead, the one with positive energy is the one trapped causing the black hole to grow and the universe to shrink?
In the popular quantum model of virtual pair loops within the Heisenberg uncertainty, close to the horizon, one has to keep in mind that virtual particles have to interact with something to become real particles which could be absorbed by the black hole. To become real there should be an interaction, and that interaction can only take energy from the black hole. From momentum conservation, if one real created particle has enough energy to leave against the gravitational field, the other will be absorbed because of its direction. The emitted particle in this frame takes away energy from the black hole and adds it to the rest of the universe.
In this link there is the paper itself where the connection with black body radiation is given. And here is another link with theoretical calculations.