I’m not sure this is what Munkres is talking about, but here’s something he *could* be talking about. Let $X$ be a completely regular space and let $C_b(X)$ be the ring of bounded continuous functions $X to mathbb{R}$. This is a commutative Banach algebra when equipped with the sup norm, and in fact it is a $C^{ast}$-algebra when equipped with the trivial involution.

Then the Gelfand spectrum of $C_b(X)$ is canonically isomorphic to $beta X$; equivalently, $C_b(X)$ is canonically isomorphic to $C(beta X)$. One application here is that by the Riesz representation theorem, positive linear functionals on $C_b(X)$ can be identified with Borel regular measures on $beta X$. A special case of this construction is described in the Wikipedia article.

Stone-Čech compactification is frequently mentioned in the book Carothers: A short course on Banach space theory, so this might be a good guess where to look for such applications.

One of the applications is Garling’s proof for Riesz representation theorem for $C(K)$, $K$ being compact. The proof is first done for the Stone-Čech compactification of discrete space and then extended to arbitrary compact Hausdorff spaces.

You can check Chapter 16 of Carother’s book, or some of the following papers:

- D. J. H. Garling: A ‘short’ proof of the Riesz representation theorem, Mathematical Proceedings of the Cambridge Philosophical Society, 1973 – Volume 73, Issue 03, pp 459-460
- D. J. H. Garling: Another ‘short’ proof of the Riesz representation theorem,

Mathematical Proceedings of the Cambridge Philosophical Society, 1986, Volume 99 / Issue 02, pp 261 – 262 - Donald G. Hartig: The Riesz Representation Theorem Revisited, The American Mathematical Monthly, Vol. 90, No. 4 (Apr., 1983), pp. 277-280
- One of these papers mentions that this proof can also be found in the book R. B. Holmes: Geometric functional analysis and its applications.

I am making this post community wiki, so that if someone is more familiar with the proof, they can add more details. (I only know that such a proof exists and more-or-less understand what are the basic ideas behind it.)