The article is a bit dodgy in this regard. It just means that at a quantum level, the energy fluctuations (as observed in this experiment) are not big enough to cause significant space-time alterations.
If the situation was chaotic (which I would presume a system with many random perturbations to be) the expectation is for the repeated small changes to have a significant impact.
That is, if the burst lasted 1 second (I made this number up!), you'd expect all photons to arrive within 1 second of each other. But if space-time foam had a strong effect, you would expect the same average time, but you might expect the spread of the data to be larger: maybe 5 seconds (also made up). (Essentially, randomness like this would be expected to increase the standard deviation of the arrival times.)
So by measuring the spread of the photon arrival times, you can put an upper bound on how big the space-time foam effects can be: the actual spread is (very roughly) the sum of the actual length of the event plus the spreading due to space-time foam. These folks are claiming that for certain models of space-time foam, the photons they observed arrived too close together to be consistent with the existence of that foam at the expected scale (assuming it's not a statistical fluke).
http://en.wikipedia.org/wiki/Brownian_motion
All those molecules bumping into a particle do cancel each other out over a long time. However, particles do drift noticeably due to this phenomenon.