As mentioned in the answers linked in the comments to the post (see, for example, the answers to An explanation of Hawking Radiation), the notion of particle does not make sense in curved spacetime (or even in flat spacetime, if you consider non-inertial observers). If you're interested in a more mathematical explanation (i.e., if you have studied QFT before), this post has an outline of the derivation without actually opening up all of the calculations.
As for the entanglement issue, notice that the gravitational field doesn't really play a role in your question. As an example, consider the following situation. someone picks a coin and cuts it in half, separating heads from tails. This person then puts each half in a closed box and gives one of the boxes to me and the other to you. They then instruct me to go to Alpha Centauri (4 light-years from here) and only then open the box.
The second I open the box, I see I have the heads side. Hence, I immediately know you have tails, even though you are four light-years away. How could I know that so fast without violating causality?
The point is that causality was never violated. There is a correlation between the sides of the coin that exists since well before I traveled to Alpha Centauri. Similarly, in Hawking radiation, there are quantum correlations in the quantum fields. Or, pictorially, between the "particles". The field outside doesn't have to know what is going on inside because the information that fell in was correlated to the information outside before falling in.
This is pretty much the meaning of entanglement. One can say that the modes that fell inside the black hole are entangled with modes outside of the black hole.
However, there is a caveat. If you only look at the information outside of the black hole, you cannot reconstruct the entire state. In other words, if you only have access to the outside of the black hole, you cannot know what is going on inside it, despite the correlations. For example, if I measure some property of a particle far away from the black hole (where the notion of particle actually has meaning), I will not be sure whether this particle is entangled to something inside the black hole or if it is not. This is why people talk about information loss upon black hole evaporation. The information that was inside the black hole simply vanishes, according to Hawking's calculation. Some physicists dispute the result, others agree with it. It is still a matter of debate whether this is actually an issue or not. If this interests you, I have discussed more about information loss in this post.