Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We consider $I^*(G)$ the set of invariant polynomials of $G$ ie the multilinear functions $f: \frak{g} \times ... \times \frak{g} \rightarrow \mathbb{R}$ satsifying
$f(Ad(g)X_1,...,Ad(g)X_k) = f(X_1,...,X_n)$
for all $g \in G$ with $Ad(g) = (R_g)_*$ the map induced on $\frak{g}$ by the right multiplication with an element of $G$. It turns out very surprisingly that
1)the form
$f(\Omega)(V_1,...,V_{2k}) = f(\Omega(V_1,V_2),...,\Omega(V_{2k-1},V_{2k}))$
can be projected to a form in $H^{2k}(M)$ is independant of the choice of connection for $P$ and
2) in the case of the $GL(n,\mathbb{K})$-bundle (with $\mathbb{K} = \mathbb{R} $ or $ \mathbb{C}$) associated to a vector bundle this form represents a characteristic class of the bundle.
My problem is that I have read proofs of the statements but I do not feel I understand why they are true. In other words, what made Chern and Weil expect that this construction would yeild characteristic classes?