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 singulat 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.