Since its use in the Lottery Ticket Hypothesis, iterative magnitude pruning (IMP) has become a popular method for extracting sparse subnetworks that can be trained to high performance. Despite its success, the mechanism that drives the success of IMP remains unclear. One possibility is that IMP is capable of extracting subnetworks with good inductive biases that facilitate performance. Supporting this idea, recent work showed that applying IMP to fully connected neural networks (FCNs) leads to the emergence of local receptive fields (RFs), a feature of mammalian visual cortex and convolutional neural networks that facilitates image processing. However, it remains unclear why IMP would uncover localized features in the first place. Inspired by results showing that training on synthetic images with highly non-Gaussian statistics (e.g., sharp edges) is sufficient to drive the emergence of local RFs in FCNs, we hypothesize that IMP iteratively increases the non-Gaussian statistics of FCN representations, creating a feedback loop that enhances localization. Here, we demonstrate first that non-Gaussian input statistics are indeed necessary for IMP to discover localized RFs. We then develop a new method for measuring the effect of individual weights on the statistics of the FCN representations ("cavity method"), which allows us to show that IMP systematically increases the non-Gaussianity of pre-activations, leading to the formation of localized RFs. Our work, which is the first to study the effect of IMP on the statistics of the representations of neural networks, sheds parsimonious light on one way in which IMP can drive the formation of strong inductive biases.