There is a new hot area of research to make black-box models interpretable, called Explainable Artificial Intelligence (XAI), if you want to gain some intuition on one such approach (called LIME), read on!
Before we dive right into it it is important to point out when and why you would need interpretability of an AI. While it might be a desirable goal in itself it is not necessary in many fields, at least not for users of an AI, e.g. with text translation, character and speech recognition it is not that important why they do what they do but simply that they work.
In other areas, like medical applications (determining whether tissue is malignant), financial applications (granting a loan to a customer) or applications in the criminal-justice system (gauging the risk of recidivism) it is of the utmost importance (and sometimes even required by law) to know why the machine arrived at its conclusions.
One approach to make AI models explainable is called LIME for Local Interpretable Model-Agnostic Explanations. There is already a lot in this name!
LIME is not trivial to use and it can be very slow, but it is a great way to visualize models.