Like tides on the ocean, it is a consequence of the tidal forces (if the claim is correct at all which I am verifying now). Tidal forces result from inhomogeneities of the gravitational acceleration.
If we approximated the external gravitational acceleration $\vec g$ caused by the Moon or the Sun to be constant all over the Earth and its vicinity, the Earth and all its parts would just "freely fall" and one couldn't feel the effect of the Sun or the Moon at all.
However, the field $\vec g$ isn't quite uniform – it points to the direction of the center of the Sun or the Moon (instead of having parallel lines); and it decreases as $1/r^2$ with the distance from the Sun or the Moon.
When the Moon is overhead, the points in your head (closer to the Moon) have a greater $\vec g$ than those in your feet. This difference tries to make your body stretch in the vertical direction, relatively speaking. This goes the sea level goes up at this point. Because of the even symmetry, the sea level also goes up if the Moon is directly underfoot. However, on the maximum circle in between these two extreme points on the Earth defined by the Moon, the effect is reverted and the sea level goes down.
These people argue that the same "stretching" acts on the atmosphere as well. It's somewhat plausible but it seems to me that the difference of the pressure will only be equivalent to the 1-meter-or-so change of the altitude that you get from the increased sea level i.e. from the deformation of the equipotential surfaces. Due to omnipresent oscillations, this change pressure by 0.01% should be almost unobservable.
Addition: the paper in Geophysical Research Letters is here:
They claim that the main effect of the tidal forces is on the relative humidity and the effect may actually be seen in the precipitation data.