Let f : C → C be a complex function.
The target copy of C is always colored in this way. For example, the unit circle of the target plane is colored gray.
But what you see on your screen is the domain copy of C, which is colored by giving each point z in the domain the same color as the point f(z) in the target.
For example, if you enter exp(z), i.e. you take f(z) = exp(z), you will be able to see that exp(z) maps the imaginary axis to the unit circle!
The size of the squares tell you how fast f(z) is changing. More precisely, the side length of the squares is reverse proportional to |f '(z)|.
If you look at Log(z), you can see that this function is discontinuous on the negative real axis because the colors jump from green to blue.
You are encouraged to examine all functions you encounter with the Complex Function Graph Viewer, this is a great way to see what is going on.
Have fun!