Basically each input gets multiplied by some weight that gets adjusted through each iteration. The product of the input and weight gets put through an activation function, and the outcome of that can be interpreted as the network's prediction of the class.
So you see the first neuron's input is just x1. You can see in the little graph at x1 that it's split down the middle with orange on one side and blue on the other. You can think of adjusting the weight on that neuron as adjusting where along the x axis the split occurs. All points on the orange side are classified orange and all on the blue side are classified blue. If you picked a data set like the spiral one or whatever, that neuron alone isn't going to make very many correct classifications. That's because it only gets the x1 value as input and can only affect the output by multiplying x1 by some weight, which would only have the affect of shifting the classification boundary left or right. You can see the same thing happening for the second neuron with input x2 except that now it splits along the y axis. Again that alone isn't going to match the data very well.
But then you get to the second layer, and the input of each neuron in the second layer is the output of each neuron in the first layer. So these neurons are able to take into consideration both x1 and x2 and are able to divide the data in more complex ways. So you can think of the neurons in each layer of the neural network as being able to consider more and more complex properties of the data in forming its output.
