Physicists stipulate that the electrons or photons in the EPR experiment have correlated properties due to being produced at the same time and place in the same particle source, and so the interactions of these particles with physically separated detectors also are correlated. This is not in dispute. What is in dispute is whether it is possible, at least in principle, to come up with a more fundamental theory than quantum mechanics, in which all the effects are propagated in the normal, intuitive way, at speeds less than or equal to the speed of light. (This is, after all, a problem about the propagation of light!)
Bell’s argument proceeds more or less as follows: (1) The correlated particles reach the detectors, where they proceed to cause correlated detector results. (2) In classical physics, Detector A is distant from Detector B and therefore has no effect on it; likewise Detector B is distant from Detector A and can have no effect on it. (3) Therefore the readings at Detector A and Detector B are ‘independent’ and we can write P(AB|background conditions) = P(A|background conditions)P(B|background conditions). (4) From this we can deduce certain mathematical expressions that are violated by actual experiments, which instead show exactly the correlations stipulated in step (1). (5) Therefore (2) is wrong, and indeed Detector A does affect Detector B, in an exact instant, across any distance whatsoever; and vice versa, in a special, quantum mechanical way. This is ‘quantum non-locality’. Furthermore it is impossible to explain the correlations as effects propagated at or below light speed, and you shouldn’t even try to do so; to do so would be like trying to make a perpetual motion machine or square the circle.
What is actually wrong, however, is step (3), which is nothing but a formal expression of one of those logical fallacies you can find catalogued on websites such as Wikipedia and Nizkor.