I have studied complex analysis in which there is a milestone theorem called argument principle.It states that if $\Omega\subset \mathbb C$ is an open connected set and $f$ be a meromorphic function on $\Omega$,and $C$ be a simple closed contour lying inside $\Omega$ with its interior contained in $\Omega$,and no poles and zeros on $C$,then $\int_C \frac{f'(z)}{f(z)}dz=2\pi i(N_0-N_\infty)$,where $N_0$ and $N_\infty$ are respectively the number of zeros and poles counting multiplicity of $f$ inside $C$.
I understand the statement but I actually want to feel the theorem.I have seen some books regarding $\frac{f'(z)}{f(z)}dz$ as the differential change $d\log f(z)$ in the value of logarithm and then they say that $\log(f(z))=\ln|f(z)|+i\text{arg}(f(z))$,so change in logarithm is essentially change in argument because the end point and beginning point are same resulting in constant modulus,so$\int_C d\log(f(z))$ corresponds to the change in argument of $f(z)$ when we move along $C$.I cannot understand why it is so.Can someone provide me some motivation for this?I do not want rigorous understanding because that will require differential forms which is too advanced for me,what I want is the main intuition.Also I need to clarify that I do not know anything about winding numbers.