I like the simple slogan: homotopical algebra is the nonlinear generalization of homological algebra. Let me assume that you value and appreciate homological algebra in the broadest sense as a fundamental, successful and highly applicable tool in many areas of math (otherwise I can't conceive of an argument that would be convincing for this question). At the coarsest level homological algebra is based on the idea of resolutions, i.e. that to perform algebraic operations on objects we should describe them in terms of objects that behave well for the given operations.

Now let's observe that homological algebra is a linear theory, in the sense that it deals with things like vector spaces, modules over a ring, and more generally objects of abelian categories. What if your interests involve more complicated objects that are not linear? for example, rings, algebras, varieties, manifolds, categories etc? philosophically it still makes sense that we have much to gain by resolving in some appropriate sense. Homotopical algebra is the language and toolkit built for this explicit purpose, and with many explicit applications. The $\infty$-language in my mind is just a very convenient and relatively friendly apparatus to understand, navigate and apply this theory.

Some key examples:

$\bullet$ Hodge theory. For me (and I assume many other algebraic geometers) the first instance of homotopical algebraic thinking I encountered was Deligne's construction of mixed Hodge structures on the cohomology of complex algebraic varieties, one of the most powerful tools in modern algebraic geometry. The idea is that the functor "de Rham cohomology" is very wonderfully behaved on smooth complex projective varieties, and most importantly carries a rich extra structure, a pure Hodge structure. We can take advantage of this for say any singular projective variety if we use the idea of resolution, in the form of a simplicial object (a convenient nonlinear version of a chain complex) --- we replace the variety by a simplicial smooth projective variety which is equivalent in the appropriate sense, in particular will produce the same measurement (cohomology). The existence of such is deep geometry (resolution of singularities) but its explicit applications don't require explicit knowledge of this geometry. It now follows that the singular variety's cohomology carries the appropriate derived version of a pure Hodge structure, namely a mixed Hodge structure.

$\bullet$ The tangent complex. Another seminal circa 1970 application is the Quillen-Illusie theory of the tangent complex. Again we want to do basic geometry - this time calculus - on a singular variety, or perhaps let's say a commutative ring, so we resolve it in the sense that befits the problem. We like affine spaces for taking derivatives etc, so if we want to calculate derivatives (tangent spaces) on a singular variety we should resolve it by such --- replace a ring by an appropriate free resolution (this time a COsimplicial variety). This gives us a way to extend the basic tools of calculus to singular varieties, with many corresponding applications.

$\bullet$ The virtual fundamental class. This is an elaboration on the previous point which is much more recent. We would like now to integrate on a class of singular varieties, so need a version of the fundamental class. The varieties in question arise as moduli spaces (say in Gromov-Witten or Donaldson-Thomas theories), which means they are relatively easy to resolve in a natural way (express as a derived moduli problem). As ordinary varieties they are very badly behaved (eg are not even equidimensional) but the derived moduli problem naturally carries a fundamental class.

$\bullet$ In representation theory the key objects of study are again nonlinear --- associative algebras (or equivalently their categories of modules). Thus to perform algebraic operations on these algebras we gain much by allowing ourselves to resolve them. As mentioned above the geometric Langlands program is one place where homotopical language is extremely useful, but one can find the same issues in studying say modular representations of finite groups (eg the theory of support varieties and stable module categories). More generally Hochschild/cyclic theory, the "calculus" of associative algebras/the fundamental invariants of noncommutative geometry, are natural applications of homotopical algebra. There are many spectacular achievements in this area, one famous one being the Deligne conjecture/Kontsevich formality/deformation quantization circle of ideas. The cobordism hypothesis, in my view one of the pinnacles of homotopical algebra, has among its many facets a vast generalization of Hochschild theory.