How is the non-local nature of quantum entanglement explained? From what I understand, Einstein tried to introduce real but hidden variables to remove the apparent non-local nature of quantum entanglement, but Bell's inequality showed local realism isn't possible.
I have read physicists believe in locality than hidden variables because experiments and intuition say so, thus I have no issue with the non-existence of hidden variables.
But how is then locality/causality preserved in the case of quantum entanglement, or more specifically in the case of two particles having opposite spin projections when the EPR pair is space-like separated?
Putting it in another way, quantum entanglement is local (because it doesn't allow superluminal information transfer) but it allows for non-classical correlations between space-like separated particles. How is this correlation explained without violating relativity? The particles don't influence each other but yet they are "correlated". How are these statements true at the same time without invoking hidden variables? How is the correlation made?
 A: Quantum mechanics is local, in the sense that it doesn't allow for superluminal interactions. This does not contradict results such as Bell inequalities or anything allowed by entanglement.
The point is that quantum mechanics allows for correlations that cannot be explained by any local classical theory. But at the same time, these correlations are such that they cannot be exploited to achieve superluminal communication. This might seem a little odd, and it arguably is, but is perfectly consistent: there exist types of correlations that are nonlocal but at the same time cannot be used to carry any kind of information.
Of course, the above is true as far as the current formalism of quantum mechanics is concerned. Theories that try to extend quantum mechanics might work differently, but there is no commonly accepted such formalism as of yet.
A: You are correct that Bell's theorem, in conjunction with accumulated experimental results over several decades, have demonstrated to a high degree of certainty that quantum reality is non-local.
The quantum equations can be reorganised in many ways, notably done by Bohm & Hiley to describe locally real particles accompanied by a "pilot wave". But in any such reformulation, nonlocality must manifest in the nature of the pilot wave (or similar) and its interactions with the particle. Otherwise it will not be able to predict the outcomes of all those experiments, and nor will it be an equivalent quantum formulation any more but a competing physical theory.
Suggestions that the universe is therefore locally "real", just because the particles are, beg the question as to what is meant by such "reality" if  the fuller reality entails the presence of nonlocal phenomena directing them.
Perhaps ironically, Bohm got philosophical and his motivation was as much to draw out the nonlocality via what he called the "implicate order" of the Universe; cutting the locally-real particles out of his pilot wave was in a sense just a by-product of his search. But even he had no concrete proposal on how the wave interacted.
You ask about the case of two [entangled] particles having opposite spin, when separated in space. The relevant properties of the space-separated pair are described by a single quantum wave equation. Any "determined-at-source" model is an example of local realism and fails Bell's test. Thus the entanglement is intrinsically nonlocal. (This was the essential demonstration of Alain Aspect's seminal experiment). But whether the subsequent measurement events may be retrocausally related remains a matter of deep debate. For example one may seek to distinguish between quantum (measurement) causality and temporal causality, allowing the apparent (classical) causal flow of events in time to be locally subjective.
Some of the issues over what "causality", "local" and "realism" mean to different people in this context are examined by Adrian Wüthrich in Locality, Causality, and Realism in the Derivation of Bell’s Inequality
