The application of deep learning has recently been proposed for the assessment of image quality in mammography. It was demonstrated in a proof-of-principle study that the proposed approach can be more efficient than currently applied automated conventional methods. However, in contrast to conventional methods, the deep learning approach has a black-box nature and, before it can be recommended for the routine use, it must be understood more thoroughly. For this purpose, we propose and apply a new explainability method: the oriented, modified integrated gradients (OMIG) method. The design of this method is inspired by the integrated gradients method but adapted considerably to the use case at hand. To further enhance this method, an upsampling technique is developed that produces high-resolution explainability maps for the downsampled data used by the deep learning approach. Comparison with established explainability methods demonstrates that the proposed approach yields substantially more expressive and informative results for our specific use case. Application of the proposed explainability approach generally confirms the validity of the considered deep learning-based mammography image quality assessment method. Specifically, it is demonstrated that the predicted image quality is based on a meaningful mapping that makes successful use of certain geometric structures of the images. In addition, the novel explainability method helps us to identify the parts of the employed phantom that have the largest impact on the predicted image quality, and to shed some light on cases in which the trained neural networks fail to work as expected. While tailored to assess a specific approach from deep learning for mammography image quality assessment, the proposed explainability method could also become relevant in other, similar deep learning applications based on high-dimensional images.