Do we know how Deep Neural Networks work? During last five years or so the machines using Deep Learning have surprised their programmers by unexplained success of AlphaGo, driving autonomous vehicles, machine translation of human languages, image analysis etc. The AI system works with appropriate answers to questions, but do we know how does it work?
 A: You can see a neural network as a function with a lot of parameters to choose. The function takes an input and gives you an output, just like a program. Choosing the parameters can be seen as "programming" the neural network.
We do know how it works in the sense that we know what the function is and how to choose the parameters so that the resulting neural network does the thing we want it to do.
The function has a nested form. This means that you can write it as functions inside a function that have more functions inside and so on. Once we have set the values of the parameters we can try to look at what the neural network is doing by considering those building blocks to understand the inner logic of the system. This is what we sometimes are unable to do and what is meant when one says that we don't know what a particular neural network is doing.
I highly recommend you to watch first this video and then this other video by 3blue1brown to understand how neural networks work. Once you do, the distinction between the two notions of understanding what they do will be clear to you. 
