I am interested in singularity theory by topology. I want to understand following results.

$f$ is a smooth map of a closed surface $M$ which has only fold points and cusps as its singularities. Suppose that a closed curve $c$ in $M$ intersects a singular set $S(f)$ transversely at a finite number of points.

Then the number of intersection points is odd if and only if $c$ is orientation reversing:

i.e., if and only if $\{w_1(M), [c]\}= 1$, where $w_1(M) \in H^1(M; Z_2)$ is the first Stiefel-Whitney class of $M$, $[c] \in H_1(M;Z_2)$ is the $Z_2$-homology class represented by $c$, and $\{,\}$ is the Kronecker product. $H^1$ is first cohomology and $H_1$ is first homology and $Z_2$ is order $2$ cyclic group.

Above statement is Thom's result which states that the Poincare dual to the $Z_2$-homology class represented by $S(f)$ coincides with $w_1(M)$.

**Question**

How Thom's result is used for above statement? I want to know in detail. However, I do know little characteristic classes.

Thank you for your considerations.