Motivating the de Rham theorem

Here is a really "trivial" application. Since a volume form (say from a Riemannian metric) for a compact manifold $M$ is clearly closed (it has top degree) and not exact (by Stoke's Theorem), it follows that the cohomology is non-trivial, so $M$ cannot be contractible.

I don't know if it is necessary to add yet another answer, but this theme is close to my heart. I'm not a historian, and I would be happy if someone corrects me here, but I have the impression that the idea of understanding a differential in terms of its periods, which would go back to Riemann at least, would have been a historical antecedent to de Rham's theorem. In other words, I don't think the theorem came out of a vacuum.

To explain what I mean by periods, suppose that $X$ is a compact Riemann surface of genus $g$. Then $H_1(X,\mathbb{Z})=\mathbb{Z}^{2g}$, with a basis of loops $\gamma_i$ constructed in the usual way. De Rham's theorem gives an isomorphism of the first de Rham space $H^1(X,\mathbb{C})\cong \mathbb{C}^{2g}$ by identifying a $1$-form $\alpha$ with its period vector $(\int_{\gamma_i}\alpha)$. Of course, the 19th century people would have been more interested in the case where $\alpha$ is holomorphic. In this case, the space of holomorphic forms injects into $H^1(X,\mathbb{C})$ (Proof: $\alpha=df$ implies that $f$ is holomorphic and therefore constant). This is why they could talk about this without explicitly defining cohomology first.

There is quite a number of surprising and deep statements that can be proven using de Rham. The examples I list are not elementary in any sense, but give a glimpse at the power of the theory. They all have in common that they employ features of the de Rham theory that are not at hand in singular theory.

Often de Rham theory is presented as simply being the quickest way to develop cohomology theory, but in my opinion this misses the point. First of all, whether the development of the theory is really simpler than singular theory is contestable, especially if you consider that you get a considerably weaker theory as long as you if you restrict your toolkit to the Eilenberg-Steenrod axioms. Secondly, the real power of de Rham theory becomes apparent when you study specific situations where you can apply different methods than that of standard homology theory. What are these specific situations? Well, I have three examples in mind, but certainly there are much more:

1.) Connections and curvature, i.e. Chern-Weil theory. This can be motivated by the Gauss-Bonnet formula, or, better, the Gauss-Bonnet-Chern theorem, equating the Euler number of a manifold with an integral of some differential form constructed from the curvature. Already the statement that this integral is an integer is pretty intriguing if you do not know about de Rham's theorem.

2.) Symmetry! If a compact group acts on the manifold, you can restrict to invariant forms. If the action is homogeneous, you are left with a finite-dimensional complex. So symmetry can be reduced to cut down the size of the de Rham complex, leading for example to the isomorphism $H^{\ast}(G) \cong (\Lambda \mathfrak{g}^{\ast})^G$ for compact $G$, which came as a real surprise to me when I saw it first. As far as I know, this is the simplest way to the real cohomology of Lie groups.

3.) Kähler metrics! The Hodge decomposition is of course even less elementary than the previous examples, but the statement that the dimension of the space of holomorphic 1-forms on a closed Riemann surface $S$ is precisely the genus (defined as the number of handles) is rather mysterious in the first place.