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.