The intuition behind "neural network dreams" is to "turn the neural network upside down" and instead of using it to compute the probability that the input image belongs to a certain class, use it to modify the input image, for fixed network weights, until the probability of it belonging to a given class is maximized. The first step is to define a loss function, which in this case could be the input of the output neuron corresponding to class Cat. We could also use the output of that neuron, i.e., the probability that the image is Cat, but empirically it has been found that the first choice results in better-looking images. Then, using automatic differentiation, you compute the gradient of the loss function with respect to the input features, i.e., to the pixels of the input image, not with respect to the weights of the neural networks as you do in backpropagation. Here the neural network weights are fixed.
For example, you could start with a random noise image, and compute the gradient of the activation of class Cat with respect to the input features. Now you add this gradient (multiplied by a suitable step size) to the input image, so that activation is increased. We call this process gradient ascent, because we're trying to maximize the loss (activation) instead than minimizing it as we do with (stochastic) gradient descent. This approach by itself doesn't work very well: it results in images which are basically high-frequency, apparently random patterns, which look nothing like a cat (see for example here). However, if we add some statistical constraints which are usually satisfied by natural images, such as the fact that neighboring pixels are strongly correlated, the results are better.
However, this still doesn't result in very "dreamy" images. Deep Dream uses a different approach: we don't select any class, and we don't start from a random noise image. Instead, we choose one of the neural network layers, and we give the neural network an input image. This time our loss is the mean of the layer output (before applying the ReLu nonlinearity, usually, because otherwise gradients for features with negative initial activations will be zero). Correspondingly, the gradient is the gradient of this mean activation with respect to the input features (the pixel values of the input images). Now gradient ascent will enhances whatever features increase the most the mean layer activation the most. In other words, it enhances the features detected by that layer. This results in "applying a theme" to your input image, where the theme depends from the selected layer. Each layer of the network deals with features at a different level of abstraction, so the complexity of features we generate depends on which layer we choose to enhance. For example, lower layers tend to produce strokes or simple ornament-like patterns, because those layers are sensitive to basic features such as edges and their orientations.
The algorithm is usually applied in an iterative fashion, first downscaling the input image to a minimum size (image shape, i.e., number of pixels in the vertical & horizontal direction), then applying gradient ascent, then upscaling to a successive size, until we get back to the original image size. The successive scales are called octaves. The DeepDream algorithm was originally implemented in Caffe, so it's useless for most practitioners today. Luckily, there are modern implementations in Keras and Tensorflow:
https://github.com/keras-team/keras/blob/master/examples/deep_dream.py (classic Keras implementation: only works with Inception v3, you need to modify the code manually to get it to work with other networks)
https://github.com/danielvarga/keras-deep-dream (slightly more general Keras implementation: the neural network model and the layer name can be passed to the script as CL arguments, without having to manually modify the code)
https://github.com/tensorflow/tensorflow/blob/master/tensorflow/examples/tutorials/deepdream/deepdream.ipynb (Tensorflow implementation, it uses the Inception architecture but I'm not sure which version)
