The experiment in question is written up in Bounds on Spectral Dispersion from Fermi-detected Gamma Ray Bursts, by Nemiroff, Connolly, Holmes, & Kostinski. They've written it in Physics Review Letter format, which means that it's only 4 pages long.
Basically, they put bounds on the dispersion relation for photon propagation by studying EM wave pulses from distant gamma ray bursts.
The popular science article that OP references is simplified to the point of being misleading. For one thing, it gives the impression that the researchers were actually looking at just 3 photons. They were dividing the photons into bunches, starting with the 3 most tightly clustered, then 4, then...
Regarding interpretation: The word 'interpretation' is not being used in the technical/philosophical sense -- "what do these variables in our physical model mean?" -- but rather in the sense of "How can I translate this from science to pop-science?". They're referencing some vague ideas of Wheeler's; he speculated that in a quantum gravity theory, space might be "foamy" or "fuzzy". This language predates essentially all technical work on quantum gravity, and is dragged out when poetic description is required. In this case, the idea is that quantum gravity effects would change the dispersion relation for photons at extremely high energies. Apparently not.