From Ars Technica:
Delayed-choice entanglement swapping consists of the following steps. (I use the same names for the fictional experimenters as in the paper for convenience, but note that they represent acts of measurement, not literal people.)
Two independent sources (labeled I and II) produce pairs photons such that their polarization states are entangled. One photon from I goes to Alice, while one photon from II is sent to Bob. The second photon from each source goes to Victor. (I'm not sure why the third party is named "Victor".)
Alice and Bob independently perform polarization measurements; no communication passes between them during the experiment—they set the orientation of their polarization filters without knowing what the other is doing.
At some time after Alice and Bob perform their measurements, Victor makes a choice (the "delayed choice" in the name). He either allows his two photons from I and II to travel on without doing anything, or he combines them so that their polarization states are entangled. A final measurement determines the polarization state of those two photons.
The results of all four measurements are then compared. If Victor did not entangle his two photons, the photons received by Alice and Bob are uncorrelated with each other: the outcome of their measurements are consistent with random chance. (This is the "entanglement swapping" portion of the name.) If Victor entangled the photons, then Alice and Bob's photons have correlated polarizations—even though they were not part of the same system and never interacted.
Now, this is rather interesting in itself. My interpretation is that the universe already "knows" whether Victor will entangle or not at the time of Alice's and Bob's measurements (since it controls Victor's random generator). This kind of avoids the paradox.
The real interesting question, however, is why they haven't designed the experiment in the following way:
Instead of letting Victor randomize whether to entangle, he should base his decision on the measurements of Alice and Bob: if they measured correlated polarizations, he should not entangle; if they measured uncorrelated polarizations, he should entangle.
Seemingly, this would force the universe to produce correlated polarizations for Alice and Bob, despite there being no entanglement-chain connecting them. (Because, clearly, it would be contradictory if they were uncorrelated, despite there being a chain connecting them.)
To me, this seems like a more interesting experiment/result. Any idea why they didn't do it this way?
Update to answer @Nathaniel's comment: I don't think several measurements is necessary. Let's say that both Alice and Bob check for horizontal polarization: then if Victor decides to entangle, both Alice and Bob must get the same outcome (either fire, or don't fire). Obviously, it is not contradictory that they both get the same outcome even if there's no chain, but the experiment I'm suggesting would imply that they would always get the same outcome, despite there never being any chain.