Attribution methods are a popular class of explainability methods that use heatmaps to depict the most important areas of an image that drive a model decision. Nevertheless, recent work has shown that these methods have limited utility in practice, presumably because they only highlight the most salient parts of an image (i.e., “where” the model looked) and do not communicate any information about “what” the model saw at those locations. In this work, we try to fill in this gap with Craft -- a novel approach to identify both “what” and “where” by generating concept-based explanations. We introduce 3 new ingredients to the automatic concept extraction literature: (i) a recursive strategy to detect and decompose concepts across layers, (ii) a novel method for a more faithful estimation of concept importance using Sobol indices, and (iii) the use of implicit differentiation to unlock Concept Attribution Maps. We conduct both human and computer vision experiments to demonstrate the benefits of the proposed approach. We show that our recursive decomposition generates meaningful and accurate concepts and that the proposed concept importance estimation technique is more faithful to the model than previous methods. When evaluating the usefulness of the method for human experimenters on the utility benchmark, we find that our approach significantly improves on two of the three test scenarios (while none of the current methods including ours help on the third). Overall, our study suggests that, while much work remains toward the development of general explainability methods that are useful in practical scenarios, the identification of meaningful concepts at the proper level of granularity yields useful and complementary information beyond that afforded by attribution methods.