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Neural networks (NNs) usually hinder any insight into the reasoning behind their predictions. We demonstrate how influence functions can unravel the black box of NN when trained to predict the phases of the one-dimensional extended spinless Fermi-Hubbard model at half-filling. Results provide strong evidence that the NN correctly learns an order parameter describing the quantum transition in this model. We demonstrate that influence functions allow to check that the network, trained to recognize known quantum phases, can predict new unknown ones within the data set. Moreover, we show they can guide physicists in understanding patterns responsible for the phase transition. This method requires no a priori knowledge on the order parameter, has no dependence on the NN's architecture or the underlying physical model, and is therefore applicable to a broad class of physical models or experimental data.