what do hidden layers mean in a neural network?
Consider a very basic example of AND, OR, NOT and XOR functions.
You may already know that a single neuron is only suitable when the problem is linearly separable.
Here in this case, AND, OR and NOT functions are linearly separable and so they can be easy handled using a single neuron.
But consider the XOR function. It is not linearly separable. So a single neuron will not be able to predict the value of XOR function.
Now, XOR function is a combination of AND, OR and NOT. Below equation is the relation between them:

a XOR b = (a AND (NOT b)) OR ((NOT a) AND b)

So, for XOR, we can use a network which contain three layers.
First layer will act as NOT function, second layer will act as AND of the output of first layer and finally the output layer will act as OR of the 2nd hidden layer.
Note: This is just a example to explain why it is needed, XOR can be implemented in various other combination of neurons.
in a standard neural network, I'm trying to understand, intuitively, what the values of a hidden layer mean in the model.
I understand the calculation steps, but I still dont know how to think about the hidden layers and how interpret the results (of the hidden layer)
So for example, given the standard MNIST datset that is used to train and predict handwritten digits between 0 to 9, a model would look like this:

An image of a handwritten digit will have 784 pixels.
Since there are 784 pixels,  there would be 784 input nodes and the value of each node is the pixel intensity(0-255)
each node branches out and these branches are the weights.
My next layer is my hidden layer, and the value of a given node in the hidden layer is the weighted sum of my input nodes (pixels*weights).
Whatever value I get, I squash it with a sigmoid function and I get a value between 0 and 1.

That number that I get from the sigmoid.  What does it represent exactly and why is it relevant?  My understanding is that if I want to build more hidden layers, I'll be using the values of my initial hidden layer, but at this point, i'm stuck as to what the values of the first hidden layer mean exactly.
Thank you!