In case I'm going from a wrong assumption here's how I understood the experiment:
The singlet electron-positron pair has spin zero, so if we measure one electron as +x (i.e. +1 on x axis) then the positron must have -x. If we measure the electron again then it still has +x (unless we did a measurement on another axis in the meantime). Since the electron can come up as +x or -x with 50% probability each but the positron always precisely mirrors the outcome, the appearance of superluminal interaction arises, violating locality.
This disturbed Einstein, hence he postulated hidden variables, i.e. that all possible measurements (i.e. whether along x, y or z axis) are already present at creation of the singlet (a position called "realism"). But Bell's theorem shows that local realism does not hold, i.e. the distribution of probabilities when measuring at various angles (and hence various superpositions of the axes) do not follow what would be expected from hidden variables.
Now the claim seems to be that since local realism doesn't work one has to sacrifice either locality or realism to keep the other. But I can't see how giving up realism does anything to ensure locality since realism was invented by Einstein precisely to get around apparent non-locality, so if we drop realism we're still going to have non-locality.
Hence I would think that Bell's theorem only disproves hidden variables (i.e. realism) but doesn't really say anything about (non-)locality - rather, the appearance of non-locality is only due to the specific interpretation of quantum mechanics (since non-locality isn't a problem with photons and e.g. considering a -x positron as the time-reversed +x electron makes it also go away for electrons).
Does that make sense?