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.