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Breast density, which is the ratio between fibroglandular tissue (FGT) and total breast volume, can be assessed qualitatively by radiologists and quantitatively by computer algorithms. These algorithms often rely on segmentation of breast and FGT volume. In this study, we propose a method to directly assess breast density on MRI, and provide interpretations of these assessments. We assessed breast density in 506 patients with breast cancer using a regression convolutional neural network (CNN). The input for the CNN were slices of breast MRI of 128 x 128 voxels, and the output was a continuous density value between 0 (fatty breast) and 1 (dense breast). We used 350 patients to train the CNN, 75 for validation, and 81 for independent testing. We investigated why the CNN came to its predicted density using Deep SHapley Additive exPlanations (SHAP). The density predicted by the CNN on the testing set was significantly correlated with the ground truth densities (N = 81 patients, Spearman's rho = 0.86, P < 0.001). When inspecting what the CNN based its predictions on, we found that voxels in FGT commonly had positive SHAP-values, voxels in fatty tissue commonly had negative SHAP-values, and voxels in non-breast tissue commonly had SHAP-values near zero. This means that the prediction of density is based on the structures we expect it to be based on, namely FGT and fatty tissue. To conclude, we presented an interpretable deep learning regression method for breast density estimation on MRI with promising results.