what do hidden layers mean in a neural network?
AFAIK, for this digit recognition case, one way to think about it is each level of the hidden layers represents the level of abstraction.
For now, imagine the neural network for digit recognition has only 3 layers which is 1 input layer, 1 hidden layer and 1 output layer.
Let's take a look at a number. To recognise that it is a number we can break the picture of the number to a few more abstract concepts such as lines, circles and arcs. If we want to recognise 6, we can first recognise the more abstract concept that is exists in the picture. for 6 it would be an arc and a circle for this example. For 8 it would be 2 circles. For 1 it would be a line.
It is the same for a neural network. We can think of layer 1 for pixels, layer 2 for recognising the abstract concept we talked earlier such as lines, circles and arcs and finally in layer 3 we determine which number it is.
Here we can see that the input goes through a series of layers from the most abstract layer to the less abstract layer (pixels -> line, circle, arcs -> number). In this example we only have 1 hidden layer but in real implementation it would be better to have more hidden layer that 1 depending on your interpretation of the neural network. Sometime we don't even have to think about what each layer represents and let the training do it fo us. That is the purpose of the training anyway.
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!