What that sentence means is that when you look at the CMB in very large patches on the sky (about the size of the moon and bigger) its temperature fluctuates from patch to patch less than we would expect.
This was already somewhat the case before the BICEP2 discovery, but BICEP2 made it much more significant. The reason for this will hopefully turn into a post of its own one day, but, essentially, the primordial gravitational waves that BICEP2 has hopefully discovered would themselves have seeded temperature anisotropies on these large angular scales. Previously, we could just assume that the primordial gravitational waves had a really small amplitude and thus didn't affect the temperature much at all. Now, however, it seems like they might be quite large and therefore, this apparent lack of power becomes much more pertinent.
That's all fine and is something that any model of inflation that hopes to explain the origin of these gravitational waves will need to explain, despite what many cosmologists already writing papers on the ArXiv seem to want to believe (links withheld). As a side, ever-so-slightly-frustrated, note, the only papers I've seen that have actually analysed the data, rather than repeating old claims, have confirmed this problem that was clear from, at the latest, the day after the announcement.
But why does it imply a "cosmological coincidence problem"? And why is it a new coincidence problem? What's the old one?
The old cosmological coincidence problem
The energy density attributable to a cosmological constant is, well, constant. The energy density of matter and radiation drops as the universe expands. Right now, today, the energy densities of matter and of "dark energy"/the cosmological constant appear to be similar. If we extrapolate into the distant future, almost all of the energy density of the universe will be dark energy and in the past almost none of it was.
The first cosmological coincidence problem is that we live and observe at precisely the time of this transition. That is supposed to seem a bit odd. When I'm in the right mind, I agree. There are a number of possible explanations for it, most of them anthropic (though some aren't - e.g. if dark energy is related to structure formation).
This is a problem well-known amongst cosmologists and something often pondered about.
The new cosmological coincidence problem(s?)
However BICEP2 suggests a new one. The universe is expanding. But light also travels at a finite speed. Therefore, as time goes on more of the universe becomes visible to us (at least so far) as we see farther and farther away. The result of this is that the largest angular scales we can currently see (the patches on the sky bigger than the moon) have only "recently" become visible. If we were around when the CMB formed, we would observe a much smaller fraction of the currently observable universe.
Maybe you're starting to see where the new coincidence problem comes from. The behaviour of the universe on all the angular scales smaller than the moon appears to be consistent and well described by one model, whereas the fluctuations on the largest angular scales appear to follow a different model.
Why do we observe this transition today?
If we were around billions of years ago we wouldn't even know that this funny behaviour had occurred, because these very large distance scales would still be "outside our horizon", we simply wouldn't be able to see them because light couldn't have brought us knowledge of them yet.
So, the new cosmological coincidence problem is that this strange behaviour is becoming visible to the universe at precisely the same time as we are here observing it. Why now (if it has to happen at all) and not much later or much earlier?
Is a potential explanation again anthropic? Did the universe need to have some minimum size in order for intelligent life to have enough time/space to evolve and we're now seeing the edge of our homogeneous, observable, patch? Whereas, in other, smaller (and more common?), patches there is no life to see this effect happen much earlier?
As a final comment, I can't help but think that there is then an obvious third coincidence problem that arises when you combine both of the others. If it is a strange coincidence that we are around just as dark energy comes to dominate the universe and it is a strange coincidence that we are around just as this funny feature in the primordial fluctuations of the universe becomes visible, then it is also a strange coincidence that dark energy comes to dominate at exactly the same time as the primordial fluctuations change their shape.
Is it all anthropic? Is the same physical mechanism that is responsible for dark energy also responsible for these large scale features? Have I just lost any chance of getting a permanent job in serious cosmology? Time will tell (at least I didn't put it on the ArXiv!)