Definition. Let $f:(I,\mathring I)\to(S^1,1)$ be a continuous function, the degree of the function $f$ is defined by $$\deg(f)=\widetilde{f}(1)$$
where $\widetilde f$ is the unique lifting of $f$ with $\widetilde f(0)=0.$
$\widetilde f$ is such that $\exp\circ\widetilde f=f.$
Could someone please explain what this definition means intuitively?
I have a lot of exercises that involve this definition and I do not see any idea on how to solve them, perhaps with the intuition of this definition would help me.
Thank you.
I did read the answer of the question What does the degree of a loop in $S^1$ means *intuitively? in which the definition is different and got the idea.