The thing you are missing is one further geometric property of the $(2n{-}1)$-form $\Pi$ that Chern constructs on the unit sphere bundle $\mathsf{S}(M)$ of the oriented $2n$-manifold $M$: The fact that the pullback of $\Pi$ to any unit sphere $\mathsf{S}_x(M)\subset T_xM$ is simply the induced volume form of $\mathsf{S}_x(M)$. 

Once one establishes this, Chern's proof of the Gauss-Bonnet theorem is straightforward:  Choose a vector field $X$ on $M$ that has isolated zeroes $z_1,\ldots, z_k\in M$, and let $$U = \frac{X}{|X|}:M\setminus\{z_1,\ldots,z_k\}\to \mathsf{S}(M)$$ be the corresponding unit vector field, defined and smooth away from the $z_i$.  Let $\epsilon>0$ be sufficiently small that the geodesic $\epsilon$-balls $B_\epsilon(z_i)$ around the $z_i$ are disjoint and smoothly embedded.  On the manifold with boundary $M_\epsilon\subset M$ that consists of $M$ with these $\epsilon$-balls removed, consider the section $U:M_\epsilon\to \mathsf{S}(M)$.  By construction/definition, $U^*\Omega = U^*(\mathrm{d}\Pi)$ is the Gauss-Bonnet integrand over $M_\epsilon$.  By Stokes' Theorem,
$$
\int_{M_{\epsilon}}U^*\Omega = \sum_{i=1}^k \int_{\partial B_\epsilon(p_i)} U^*\Pi.
$$ 
Now let $\epsilon$ go to zero.  The left-hand side converges to the Gauss-Bonnet integrand over all of $M$ while the $i$-th summand on the right-hand side converges to the index of $X$ at $z_i$. (This is because $U^*\Pi$ on $\partial B_\epsilon(z_i)$ differs by a term vanishing with $\epsilon$ from the pullback of the unit volume form of $\mathsf{S}_{z_i}(M)$ to $\partial B_\epsilon(z_i)\simeq \mathsf{S}_{z_i}(M)$ under the indicial mapping induced by $U$ at $z_i$, whose degree is, by definition, the index of $X$ at $z_i$.)  Thus, passing to the limit and using the Poincaré-Hopf theorem (that the sum of the indices of the vector field $X$ is equal to $\chi(M)$), one obtains Chern's proof of the Gauss-Bonnet Theorem.

As to *why* $\Pi$ pulls back to each $\mathsf{S}_{z_i}(M)$ to be the unit volume form, you need to look at Chern's definition of $\Pi$, which uses the Pfaffian, particularly its algebraic properties.  This comes out of the computation that Chern does, and it is essentially a geometric fact, but it amounts to an explicit formula for the transgression operator defined in Chern-Weil theory for the Euler class.  Another way to look at it would be to look at the generalized Gauss-Bonnet formula, a discussion of which you can find at the MO question https://mathoverflow.net/questions/84521/a-question-on-generalized-gauss-bonnet-theorem/84960#84960.