Galaxy appearances reveal the physics of how they formed and evolved. Machine learning models can now exploit galaxies' information-rich morphologies to predict physical properties directly from image cutouts. Learning the relationship between pixel-level features and galaxy properties is essential for building a physical understanding of galaxy evolution, but we are still unable to explicate the details of how deep neural networks represent image features. To address this lack of interpretability, we present a novel neural network architecture called a Sparse Feature Network (SFNet). SFNets produce interpretable features that can be linearly combined in order to estimate galaxy properties like optical emission line ratios or gas-phase metallicity. We find that SFNets do not sacrifice accuracy in order to gain interpretability, and that they perform comparably well to cutting-edge models on astronomical machine learning tasks. Our novel approach is valuable for finding physical patterns in large datasets and helping astronomers interpret machine learning results.