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The atmosphere is chaotic. This fundamental property of the climate system makes forecasting weather incredibly challenging: it's impossible to expect weather models to ever provide perfect predictions of the Earth system beyond timescales of approximately 2 weeks. Instead, atmospheric scientists look for specific states of the climate system that lead to more predictable behaviour than others. Here, we demonstrate how neural networks can be used, not only to leverage these states to make skillful predictions, but moreover to identify the climatic conditions that lead to enhanced predictability. Furthermore, we employ a neural network interpretability method called ``layer-wise relevance propagation'' to create heatmaps of the regions in the input most relevant for a network's output. For Earth scientists, these relevant regions for the neural network's prediction are by far the most important product of our study: they provide scientific insight into the physical mechanisms that lead to enhanced weather predictability. While we demonstrate our approach for the atmospheric science domain, this methodology is applicable to a large range of geoscientific problems.