Not sure if it tells us more about why the network works, but they sure as heck get rid of a lot of connections.
Most neural network models assume that all neurons in one level are fully connected to all neurons in the next level. This leads to confusion about how ANNs work.
From my research I'd argue that MOST interactions in these networks are spurious. Once you remove them it reveals the (visual) topology (circuit diagram) that's driving the function of that network.
In the paper I wrote, I evolved gene regulatory networks (of ANNs have the same mathematical representation) such that interactions between any two nodes could be deleted, created (if W_ij = 0), or modified according to probabilities of deletion, creation, and modification. Given these probabilities, you can calculate the number of interactions that should result when the network reaches equilibrium, however what I found was that the network evolves less interactions than you would expect from the equilibrium calculation. This says that all things being equal, a network is paying a price for spurious interactions and that these will be removed in an evolutionary environment. Basically, each interaction needs to pay it's way otherwise it leads to unnecessary complexity that reduced the fitness of the network.
Let's say a specific problem has only one very specific set of connections that matters. You'll eventually add up with weights that reflects that, but that doesn't prevent a lot of other connections from having weights set during training, but that may end up being cancelled out or reduced enough to have no meaningful impact whatsoever on the end result.