Set theory is an extremely convenient language for being able to rigorously define and manipulate various "completed infinities" - not only just infinite sets such as the natural numbers or real numbers, but much larger completed infinities, such as Stone-Cech compactifications, the hyperreals, or ultrafilters, that typically need some fairly powerful set-theoretic tools, such as Zorn's lemma, to construct. One can often get by in applications using various "incomplete" or "finitary" substitutes for these objects, which require less set-theoretic machinery to set up (e.g. one may be able to largely avoid use of the axiom of choice), but the mathematics can become much messier when doing so.

Once one has set up a non-trivial amount of mathematics in the realm of infinite or continuous spaces, one can often derive finitary consequences (at least at a qualitative level) by using further tools such as compactness arguments or nonstandard analysis, which again are most easily discussed if one is working within a set theoretic framework. A good example of this is the Furstenberg correspondence principle that allows one to derive combinatorial statements about finite sets of integers using the infinitary language of ergodic theory, which can require a non-trivial amount of set theory to work with (e.g. when using tools such as disintegration of measures with respect to a sigma algebra). I am personally fond of using the technique of ultrafilters (or nonstandard analysis) as a bridge between the finitary world of "practical" mathematics and the infinitary world described by set theory, as discussed for instance in this blog post of mine.

(One important caveat though: if one directly uses tools such as ultrafilters or compactness to transfer infinitary results to finitary results, one often ends up with conclusions that are qualitative in nature, or quantitative only with extremely poor explicit bounds. Often, additional effort is then required to obtain quantitative finitary results with bounds that are effective enough to be useful in real-world applications. Nevertheless, the infinitary results can show the way forward, and serve as an excellent source of analogy and intuition to then develop a satisfactory quantitative finitary theory.)

Set theory is an extremely convenient language for being able to rigorously define and manipulate various "completed infinities" - not only just infinite sets such as the natural numbers or real numbers, but much "larger" completed infinities, such as Stone-Cech compactifications, the hyperreals, or ultrafilters, that typically need some fairly powerful set-theoretic tools, such as Zorn's lemma, to construct. One can often get by in applications using various "incomplete" and/or "finitary" substitutes for these objects, which require less set-theoretic machinery to set up (e.g. one may be able to largely avoid use of the axiom of choice), but the mathematics can become much messier when doing so.

Once one has set up a non-trivial amount of mathematics in the realm of infinite or continuous spaces, one can often derive finitary consequences (at least at a qualitative level) by using further tools such as compactness arguments or nonstandard analysis, which again are most easily discussed if one is working within a set theoretic framework. A good example of this is the Furstenberg correspondence principle that allows one to derive combinatorial statements about finite sets of integers using the infinitary language of ergodic theory, which can require a non-trivial amount of set theory to work with (e.g. when using tools such as disintegration of measures with respect to a sigma algebra). I am personally fond of using the technique of ultrafilters (or nonstandard analysis) as a bridge between the finitary world of "practical" mathematics and the infinitary world described by set theory, as discussed for instance in this blog post of mine.

(One important caveat though: if one directly uses tools such as ultrafilters or compactness to transfer infinitary results to finitary results, one often ends up with conclusions that are qualitative in nature, or quantitative only with extremely poor explicit bounds. Often, additional effort is then required to obtain quantitative finitary results with bounds that are effective enough to be useful in real-world applications. Nevertheless, the infinitary results can show the way forward, and serve as an excellent source of analogy and intuition to then develop a satisfactory quantitative finitary theory.)