what do hidden layers mean in a neural network?
A hidden layer in a neural network may be understood as a layer that is neither an input nor an output, but instead is an intermediate step in the network's computation.
In your MNIST case, the network's state in the hidden layer is a processed version of the inputs, a reduction from full digits to abstract information about those digits.
This idea extends to all other hidden layer cases you'll encounter in machine learning -- a second hidden layer is an even more abstract version of the input data, a recurrent neural network's hidden layer is an interpretation of the inputs that happens to collect information over time, or the hidden state in a convolutional neural network is an interpreted version of the input with certain features isolated through the process of convolution.
To reiterate, a hidden layer is an intermediate step in your neural network's process.  The information in that layer is an abstraction of the input, and holds information required to solve the problem at the output.
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!