If I am correct, you do not understand how AI works/learns and in particular what this library is about. Before I start, let me clarify something: Artificial Intelligence is not the same with Machine Learning(ML). Artificial intelligence(AI), is basically the concept of using computers that can imitate human intelligence whereas Machine Learning is a subset of AI covering machines that can learn on their own without needing explicit programming to do so.
Now, there is not one Machine Learning approach, but we generally divide ML approaches into three categories. Neural Networks, are in the category of Supervised Learning (you can search more about it). I will not discuss what Neural Networks are, as there are plenty of articles out there, but I will give you some basic examples so you can have a basic understanding of how this library does work.
Ex1) Some neural networks elements are the following: the Input Layer( a layer that accepts input features and passes on the information to the hidden layer), the Hidden Layer( it performs all sort of computation on the features entered through the input layer and transfer the result to the output layer) and the Output Layer( a layer that brings up the information learned by the network to the outer world).
Ex2) Neural Networks have an activation function that as the name implies, decides whether a neuron should be activated or not by calculating weighted sum and further adding bias with it.
Now, these two examples as you understand, are some key elements of neural networks. What does this mean? That every time you want to use for some reason Neural Networks, you will have to code them. Now, what the Library does, is to have these key elements ready in advance, so you do not spend time re-coding them. Note: The Library has much more features than these, I just want to give some examples to you. In fact, this Library uses and some other algorithms, like the GA.
Also, I see that you are asking how to use these math functions and how the AI(?) would learn. In order to understand this, you need to first understand what a neural network is and how it can be used. You can see this pdf An Introduction to Neural Networks (ed.ac.uk) if you are interested in learning more about this subject.