Analytic Applications of StoneČech compactification
Solution 1
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.
Solution 2
StoneČech 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 StoneČech 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 459460
 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. 277280
 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 moreorless understand what are the basic ideas behind it.)
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Semisimplesteps over 1 year
Following Bredon's Topology and Geometry, we let $\mathcal{F}$ be the set of all continuous maps $f:X \to [0,1]$ on a completely regular space $X$, define $X \xrightarrow{\Phi} [0,1]^{\mathcal{F}}$ by setting $\Phi(x)(f)=f(x)$ for each $x \in X$ and $f \in \mathcal{F}$, and declare the closure $\beta(X)$ of $\Phi(X)$ in $[0,1]^{\mathcal{F}}$ to be the StoneČech compactification of $X$. Munkres writes in Topology that there are "a number of applications [of the StoneČech compactification] in modern analysis", which is purportedly "outside the scope of [the] book." I looked at a few sources, including The StoneČech compactification by Russell Walker, and failed to find any application of the theorem that is overtly analytic. Perhaps I do not have sufficient background in functional analysisthis is where, I assume, the prototypical analytic applications would be into recognize the functionalanalytic applications I have encountered, but the point remains that I have yet to see such an example. So:
What are prototypical applications of the StoneČech compactification in mathematical analysis, and where can I read about them?

t.b. about 12 yearsI think the most important analytic applications of ultrafilters are ultralimits and Banach limits (in guise of invariant means on amenable groups).


Martin Sleziak about 12 yearsI've added link to the wikipedia article. (I hope this is the one you had in mind.)

Martin Sleziak over 3 yearsThe proof of Riesz theorem is mentioned also here: Proofs of the Riesz–Markov–Kakutani representation theorem.