What is the "foamy space" hypothesis that has been debunked recently? In "Space-Time Is Smooth, Not Foamy", a Space.com article, it is stated:

In his general theory of relativity, Einstein described space-time as fundamentally smooth, warping only under the strain of energy and matter. Some quantum-theory interpretations disagree, however, viewing space-time as being composed of a froth of minute particles that constantly pop into and out of existence.

What are these "quantum-theory interpretations" the article is talking about? Can you quickly summarize the physics behind this experiment?
 A: It is not possible to find the paper on the web, just the description. Quantum foam is an effort to tie up the quantum regime with the classical General Relativity. Loop quantum gravity is an effort on these lines :

The main output of the theory is a physical picture of space where space is granular. The granularity is a direct consequence of the quantization

What has been announced in this conference is one specific observation that sets the probability of granularity of space at very low level.

A team of researchers came to this conclusion after tracing the long journey three photons took through intergalactic space. The photons were blasted out by an intense explosion known as a gamma-ray burst about 7 billion light-years from Earth. They finally barreled into the detectors of NASA's Fermi Gamma-ray Space Telescope in May 2009, arriving just a millisecond apart.
Their dead-heat finish strongly supports the Einsteinian view of space-time, researchers said. The wavelengths of gamma-ray burst photons are so small that they should be able to interact with the even tinier "bubbles" in the quantum theorists' proposed space-time foam.
If this foam indeed exists, the three protons should have been knocked around a bit during their epic voyage. In such a scenario, the chances of all three reaching the Fermi telescope at virtually the same time are very low, researchers said.

(italics mine).
Of course this is not a definitive experiment and has to be repeated many times.  An opposite conclusion is taken by another astronomy group on different data.

The MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov) telescopes have detected that among gamma-ray photons arriving from the blazar Markarian 501, some photons at different energy levels arrived at different times, suggesting that some of the photons had moved more slowly and thus contradicting the theory of general relativity's notion of the speed of light being constant, a discrepancy which could be explained by the irregularity of quantum foam.

Italics mine.
A: 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.  
