TITLE: What non-categorical applications are there of homotopical algebra?
QUESTION [141 upvotes]: (To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.) 
More and more lately I have been unexpectedly running into things that might be named 'homotopical algebra' - in glancing at symplectic topology I run into Fukaya categories, and thus of $A_\infty$ categories; in looking at algebraic topology at things above the introductory level I see language like 'model categories' or simplicial sets, or more abstractly, algebraic geometry might lead me to stuff like the work of Jacob Lurie. These things feel bizarre and out-of-place to me - it's all so abstract and dry it's difficult to get a handle on, and feels so removed from ""reality"" (a.k.a., things I have a geometric intuition for).
I understand that this is a result of my maturity level - I'm only just coming to terms with categorical language, and a few years ago I would have probably made a poor joke about "abstract nonsense". Similarly, I have only recently come to love the language of cohomology and sheaves, and this is perhaps because of the problems they solve rather than the language itself; classifying line bundles on a Riemann surface, say, or any of the classical theorems of algebraic topology - both of which can be done elegantly with tools more modern than those questions themselves. I would perhaps feel more at ease if I saw that homotopical algebra can be used for similar purposes, so:
What are some down-to-earth (interesting to non-category theorists, say, that are naturally stated in non-categorical language) questions that have been solved using these sorts of tools, or such that these sorts of tools are expected to be useful? Why is the language of homotopy theory useful to the modern mathematician who is not already immersed in it?

REPLY [10 votes]: This answer has a small overlap with Niles's reply. 
I don't have a non-categorical application of all the topics you discuss but I can give you one. 
There is a topological operad called "the splicing operad" that acts on embedding spaces.  It acts on a space closely related to the space of smooth embeddings of $S^1$ in $S^3$, I call this space $\mathcal K_{3,1}$, or the space of `long embeddings' of the real numbers in Euclidean 3-space.   It turns out this operad is closely related with many of the biggest structure theorems in 3-manifold topology. Namely, 
The splicing operad (in dimension 3) is a free product of a certain free operad and $O_2$ semi-direct product the $2$-cubes operad.  Moreover, the knot space $\mathcal K_{3,1}$ is freely generated by the splicing operad and the subspace of $\mathcal K_{3,1}$ consisting of torus and hyperbolic knots. 
This theorem, on the level of the path-components of $\mathcal K_{3,1}$ is precisely a description of the geometrization theorem, restricted to the class of knot exteriors.  For the higher homotopy groups the 3-manifold content of this theorem consists of the Smale conjecture, together with the equivariant connect-sum and torus decompositions of 3-manifolds, and the theorem that the actions of finite groups on $S^3$ are conjugate to linear actions. 
So this operad sees quite deeply into 3-manifold topology.  There are analogous operads in higher dimensions, and also operads of this type for spaces of string links (due to Burke and Koytcheff).  What these operads say about embeddings and diffeomorphisms groups of high-dimensional manifolds is an open problem.  High-dimensional diffeomorphism groups are presently little-understood.