I'll convert my comments to an answer.

Characteristic classes were originally defined as obstruction classes, going back to the work of Whitney and Stiefel. The top Chern class of a complex bundle is the obstruction to an everywhere non-zero vector field, and frequently called the Euler class.

The reason for the name "Euler class" comes from the case where you're looking at the tangent bundle. When you pair this class against the fundamental class of the manifold, via Poincare duality you can interpret this as the signed intersection number of the 0-section of the tangent bundle with (a transverse perturbation of) itself. Since this number does not depend on the perturbation, there are various ways to compute it and check it's the classical Euler characteristic of the base manifold. I like to use a triangulation, and adapting a vector field to it so that the zeros of the vector field correspond to the (barycentres) of the cells of the triangulation.

The core geometric argument here is the Poincare-Hopf index theorem.

Perhaps there are more adapted ways to get the result in the context you like (I don't know), but I think this is the historical route to the result.

allthe time with this kind of stuff. Seriously though, thanks a bunch. $\endgroup$