Artificial neural networks (ANN) are computing systems vaguely inspired by the biological neural networks that constitute animal brains, how do they relate to AI?
I think you mean: "What did biology to inspire AI?". Well, the backpropagation algorithm was a great breakthrough that is somewhat inspired by our neurons giving electric impulses forward and sometimes recieving backwards, we don't know if our backpropagation method is perfect, but we do know that it's the way to do it. Secondly, neurons have dentrites that connect one to many neurons, you can say that this could also be a hint for NN architecture. Anyways, I don't know if you wanted to know this, you haven't been really specific in your question.
Artificial Neural Networks or ANNs are mathematical models of a human brain. I'm pretty sure if you search ANNs on google you will see a picture of real neurons and how they work, so that serves as a motivation to mimic the behavior of a real brain to produce complex behaviors that would not be otherwise possible with different mathematical models.
And to answer how they relate to AI, the easy to understand answer would be that, since our brains give us intelligence, we would expect the same to happen when using Artificial Neural Networks to give us Artificial Intelligence.
Another way to understand how AI and ANNs relate I think is to understand that ANNs have weight matrices and depending on the ANN architecture, there might be hundreds and sometimes even thousands of weights. All these weights (sometimes called parameters) a huge solution space is made and there exist multiple solutions to a single problem. In other words, for all these weights there are multiple possible values and for each value there exist multiple combinations and each combination is unique hence making it very diverse. This is the reason why ANNs are so effective. The huge solution space always produces a good result and hence it proves to be a great candidate for AI implementations.
You can watch the video below to see how ANNs can help in creating AIs that are capable of pretty simple but impressive things. https://www.youtube.com/watch?v=YyPdSkN26WY