It helps to go to the original article for more guidance:
Hill, S. L., Wang, Y., Riachi, I., Schürmann, F., & Markram, H. (2012). Statistical connectivity provides a sufficient foundation for specific functional connectivity in neocortical neural microcircuits. Proceedings of the National Academy of Sciences, 109(42), E2885-E2894.
The point of this paper is that they are looking at the statistics at which neurons make connections with each other in neocortex. Neocortex is a layered structure, and cells of a particular type are identified by the location of their soma as well as the structure of their dendrites and axons. A layer 2/3 pyramidal cell, for example, has basal dendrites that spread a lot in layer 2/3 and creep into layer 4, and an apical dendrite that extends and branches in layer 1. Therefore, it can potentially receive inputs from any cell that has axons in layers 1 -> 4.
One hypothesis is that the primary determinant of which cell types are connected to which is simply the morphology of each cell type. If one cell Type A has a bunch of dendrites in a given layer, let's say layer 3, and another cell Type B has a bunch of axons in layer 3, they will have a lot of opportunities to contact and make synapses.
A second hypothesis is that cells connect with specialized guidance cues. For example, dendrites of Cell Type A release a peptide BALover that is detected by Cell Type B, and causes Cell Type B's axons to grow toward dendrites of Cell Type A. In this system, you could still get lots of dendrites of Cell A in one layer and lots of axons of Cell B in the same layer, but caused by the specific guidance cues from A->B.
In the second situation, however, we would expect the arrangement of axons and dendrites within that layer to be non-random. That is, we would find processes of B in greatest density nearest to processes of A.
If you labeled a bunch of Cell Type A from one animal, and a bunch of Cell Type B from another animal, and overlay them, they should have just as many contacts as if you labeled Cell Type A and B in the same animal if the connections are random due to morphology. If, however, there are specific guidance cues involved, you would find fewer.
Hill et al. 2012 shows evidence in favor of the first hypothesis: they get about as many contacts if they take cells from two individuals and overlay them. This is what they mean by "random placement": not completely random in all dimensions but rather random within a layer in cortex, as if the cells are taken from two different animals. Certainly guidance cues are still used and are still important for setting up the structure of cortex, but it seems like individual cells are mostly just programmed to take up a certain shape, and that shape determines their rough probability of connections. This is a good thing for the Blue Brain project, because it means that they can use composite data from many animals to establish connectivity patterns.