This video demonstrates some simple examples of the process of mapping in the complex plane. 
 
The complex variable s, on the left-side graph, will traverse a circular path with radius 5, in a clockwise direction.  

The right-side graph will show the path traversed by a function, L(s).  
This is called the mapping of s onto L(s).  
For this simple mapping L(s)=s.
Since L(s)=s, then L(s) has a zero at the origin.  

This is shown on the left side graph.  
As you can see the variable s, on the left is traversing a circular path.
Shown in pink on the left side graph is the angle subtended by a line from the zero to the circular path. 
  
The right side path shows, in pink the angle subtended by a line from the origin to the circular path. If this angle had been counterclockwise, it would be blue instead of pink.
 
Notice that as we encircle the zero in s, we also encircle the origin in L(s), both in a clockwise direction.

Now we move the location of the zero to s=-4, so L(s)=s+4.  
Now the path in L(s) is offset by 4.  
However, we see that the path in L(s) still encircles the origin once in a clockwise direction.

If we move the location of the zero even furtherto s=-6, so L(s)+6, the path in L(s) is now offset by 6.  
However since we no longer encircle the zero in s, we no longer encircle the origin in L(s).

The point of all of this will become clear later.  The take-home concept is that by looking at the plot in L(s) we can tell if the path in s encircles any zeros.  We don't know where the circles are, but it turn out that that isn't important for us.