This is not really a question related much to biological neural networks, rather it's a concept from artificial neural networks that are arranged in layered structure.
In a layered artificial neural network, "neurons" of one layer receive input only from the previous layer, and give output only to the subsequent layer. This creates a computationally efficient feed-forward structure: there's no complicated recurrence, once you know the activities of one layer you can calculate the activities of the next layer by a single multiplication step between the weight matrix and activity vector of the previous layer, followed by a sum for each unit.
From the README of the software you link to:
There are 3 main levels of structure: Network, Layer and Prjn (projection). The network calls methods on its Layers, and Layers iterate over both Neuron data structures (which have only a minimal set of methods) and the Prjns, to implement the relevant computations. The Prjn fully manages everything about a projection of connectivity between two layers, including the full list of Syanpse elements in the connection. ... The Layer also has a set of Pool elements, one for each level at which inhibition is computed
So, while the neurons in this network may have some biological aspects like integration over time, it seems they are arranged into layers like a classical ANN, information is shared between layers through a "Prjn" object that manages the "synapses" between two layers. Additionally, each layer seems to have its own inhibitory unit. I assume the way this is modeled is that every neuron in a layer gives input to an inhibitory unit, which then reduces the activity within that layer as sort of a gain control. Again, while they may be biologically motivated, these are all artificial constructions for a computational model chosen by the authors.
While you could conceivably still map these connections all into an NxN matrix, everything would be zero except for upper triangular matrices corresponding to each pair of layers. Nothing about hierarchical organization says you can't put it in a larger matrix, just that it doesn't necessarily make sense to do so. For example, you could put students in a classroom into a longer list by school, that doesn't mean there isn't an underlying organization of students within classroom just because you abstracted out that level.
While there are also layered structures in biological networks (like neocortex) these layers do not function like artificial neural network layers, as neurons within layers are strongly connected to each other, and while there is some evidence for a "feed-forward" pathway (roughly, layer 4 to 2/3 to 5 to 6), this is only a moderate bias in connectivity and only a minority of connections actually follow this canonical pattern.
