the net change in argument
Picture standing at the origin and watching a moving point f(z) as z travels once around a loop. You only track the angle to that point — where it sits on the clock. By the time z returns home, the angle may have wound forward, backward, or netted to zero. The net change in argument is the total angle swept, counting full revolutions with sign. Dividing it by 2 pi gives a whole number of net turns: the winding number of the image curve about 0.
Formally, write f along the contour as f = |f| e^(i theta) with theta = arg f. As z traverses gamma once, the modulus |f| returns to its start so the log-modulus change is zero around a closed loop, but arg f need not return to the same branch — it can have advanced by an integer multiple of 2 pi. That integer is the heart of the argument principle: (1 / (2 pi i)) times the integral over gamma of f'/f dz has imaginary part equal to (net change in arg f) / (2 pi), and this equals (zeros) minus (poles) inside. So 'how many net times does the pointer spin?' and 'how many zeros minus poles?' are literally the same number. You can even compute it by hand on a coarse loop: track arg f at sample points, add up the small angle steps, never letting a step jump by more than pi, and read off the total.
This is the most visual face of the whole counting machinery, and it is what makes graphical and numerical methods practical — software plots the image curve f(gamma) and literally counts how many times it circles the origin. It explains the Nyquist stability criterion (count encirclements of a critical point) and gives intuition for why deforming f without crossing 0 cannot change the count. The discipline to respect: arg f is multivalued, so you must follow it continuously along the path; if you reset it to the principal value at each step you will miscount the turns.
For f(z) = z and gamma the unit circle, arg f = arg z runs from 0 up to 2 pi over one loop, a net change of 2 pi, so the winding number is 1 — matching the single zero of z at the origin.
Net change in arg f over a loop, divided by 2 pi, is the winding number and equals zeros minus poles.
The change in arg is meaningful only as a continuous total along the path; resetting arg to its principal value at each point destroys the count. The net change of log|f| around a closed loop, by contrast, is always zero.