Generating human-readable explanations from a machine learning algorithm or NN is probably an impossible problem to solve in full generality.
However, for many particular systems, doing so should be feasible.
Also it should be possible to train the NN model to explain its decision, even in plain language.
I think the "right to explanation" law is pretty sensible, people should have a right to question the algorithmic judgment that impacts their life. It will be somewhat inconvenient for big companies applying ML to make these decisions, but it won't ruin their business.