The result you're thinking of is the following:
Proposition. Let $M$ be a smooth manifold, and let $\mathscr{C}^\infty_M$ be the sheaf of smooth functions on $M$.
For any positive integer $n$, there is a natural bijection between isomorphism classes of $n$-dimensional smooth vector bundles on $M$ and principal $\mathrm{GL}(n, \mathscr{C}^\infty_M)$-bundles on $M$, or equivalently, smooth principal $\mathrm{GL}(n, \mathbb{R})$-bundles on $M$.
For any sheaf of groups $\mathscr{G}$, there is a natural bijection between isomorphism classes of principal $\mathscr{G}$-bundles and elements of the Čech cohomology group $H^1 (M, \mathscr{G})$.
Putting the two bijections together gives us what we want.
The connection between the three ideas is more-or-less straightforward, even if the verification of the details is tedious. First of all, suppose $E$ is a $n$-dimensional smooth vector bundle on $M$. Then, there is an open cover $\mathfrak{U}$ of $M$ with respect to which $E$ is trivial, and – fixing a local trivialisation – the transition maps give us a Čech 1-cocycle for the sheaf $\mathrm{GL}(n, \mathscr{C}^\infty_M)$ with respect to the cover $\mathfrak{U}$. A Čech 1-coboundary for a sheaf is the same thing as a global section, and it is not hard to check that global sections of $\mathscr{G} = \mathrm{GL}(n, \mathscr{C}^\infty_M)$ act on the set of local trivialisations of $E$. Moreover, any two trivialisations of $E$ over $\mathfrak{U}$ are related by such a global section. Thus, we obtain a "characteristic" element of $H^1(\mathfrak{U}, \mathscr{G})$ for each trivialising cover $\mathfrak{U}$ – but we're not done yet.
Suppose we have another trivialising cover $\mathfrak{V}$; then we get an element of $H^1(\mathfrak{V}, \mathscr{G})$. But $H^1(\mathfrak{U}, \mathscr{G})$ and $H^1(\mathfrak{V}, \mathscr{G})$ are different groups, so how do we compare the characteristic elements we get? Well, first of all, we can take a common refinement $\mathfrak{W}$ of $\mathfrak{U}$ and $\mathfrak{V}$, and a trivialisation over $\mathfrak{U}$ or $\mathfrak{V}$ induces a trivialisation over $\mathfrak{W}$ by restriction. Similarly, cocycles and coboundaries can be refined, so we get homomorphisms
$$H^1(\mathfrak{U}, \mathscr{G}) \rightarrow H^1(\mathfrak{W}, \mathscr{G}) \leftarrow H^1(\mathfrak{V}, \mathscr{G})$$
and it turns out that the two characteristic elements we got become equal in $H^1(\mathfrak{W}, \mathscr{G})$ under these homomorphisms. Thus we can pass all the way to $H^1(M, \mathscr{G})$ – and the characteristic element we get there only depends on the isomorphism class of $E$.
In the reverse direction, we must find a vector bundle whose characteristic element is some chosen element of $H^1(M, \mathscr{G})$. Since $H^1(M, \mathscr{G})$ is defined as the filtered colimit of $H^1(\mathfrak{U}, \mathscr{G})$ over all the open covers $\mathfrak{U}$, we can pick an open cover $\mathfrak{U}$ and a Čech 1-cocycle w.r.t. $\mathfrak{U}$ that represents the chosen cohomology class. A standard argument then shows that we can construct a vector bundle $E$ which is trivialised over $\mathfrak{U}$ with transition maps given by the chosen cocycle.
Almost exactly the same argument shows that there is a bijection between elements of $H^1(M, \mathscr{G})$ and isomorphism classes of principal $\mathscr{G}$-bundles.
I'm afraid I don't actually know of a reference for this situation exactly. Milne talks about the second bijection in §11 of his Lectures on étale cohomology, and Johnstone more-or-less defines $H^1(M, \mathscr{G})$ using the bijection in §8.3 of Topos theory.