I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is From Calculus to Cohomology by Madsen & Tornehave. I know the statement of the theorem is as follows:
Let $M$ be an even-dimensional compact, oriented smooth manifold, $F^{∇}$ be the curvature of the connection $∇$ on a smooth vector bundle $E$.
$\int_M Pf \left( \frac{−F^\nabla}{2 \pi} \right) = \chi(M^{2n}) .$
My questions are: how does this relate to counting (with multiplicities) the number of zeros of generic sections of the vector bundle? Also, are there other good references for learning this topic? Thanks.