I'm not an expert and you most probably would have found the intuition behind using ReLU already. However, this was an interesting post and I'd like to share my thoughts. :)
Why do we need non-linear activation functions?
A neuron computes the linear function, $\small z = w^Tx + b $. Let's suppose we do not have a non-linear activation function.
Feed-forward layers without non-linear activation functions
The neuron in its succeeding layer will be computing the same feature, and just scales up (or down) the magnitude of the feature.
Even if we add a third or 4th layer, the model learns nothing new, it keeps computing the same line it started with.
However, if we add a slight non-linearity by using a non-linear activation function, for e.g. ReLU, $\small g(z) = max(0, z)$, then the neuron in the succeeding layer will be able to compute a new feature (a different line).
Feed-forward layers with a non-linear activation function (eg. ReLU)
Now, the model can actually learn something new, rather than get stuck at computing the same feature over and over.
If we add a third layer (in the 2nd image), the model will be able to learn a feature with 4 sides (quadrilateral), and so on.