Here is an example of level curves that i got from a site:
This is a topographical map of some mountain. The contours are the level curves i.e. they are the curves of constant elevation. If f(x,y) = height of mountain at point (x,y), f(x) = c are the level curves of height c.
It is similar for a complex number, except that you are mapping 2 dimensions onto 2 dimension whereas in the above example you are mapping 2 dimensions to 1 dimension. It is somewhat analogous.