I think, they have an intrinsic appeal: If one cares about manifolds, one should care about maps between manifolds. As maps between spaces appear so often in topology, of course also homotopy groups do. But some concrete examples:

1) Thom related the study of bordism classes of manifolds to the (higher) homotopy groups of Thom spaces. [Later this was rephrased into homotopy groups of Thom spectra.] This allowed him to use methods from algebraic topology to completely classify manifolds up to bordism.

1b) In particular, framed bordism classes are in relation to (stable) homotopy groups of spheres. These in turn are, by Kervaire-Milnor, in close relation with exotic differentiable structures on spheres.

2) The elements of the homotopy groups of $SO(n)$ are in one-to-one correspondence with isomorphism classes of vector bundles on spheres.

3) An example for the answers of John and Ryan: The Poincare conjecture tells us that every closed manifold homotopy equivalent to the sphere is actually homeomorphic to the sphere. How to test whether a manifold $M$ is homotopy equivalent to the $n$-sphere? It is enough to show that $\pi_1M = 0$ and $M$ has the homology of the $n$-sphere. Indeed, by the Hurewicz theorem it follows that $\pi_nM = \mathbb{Z}$. Thus, we can find a map $S^n\to M$ inducing an isomorphism on homology. Thus, it is a homotopy equivalence (by a version of the Whitehead theorem).

4) It is very useful to identify a space as an Eilenberg-MacLane space. An Eilenberg-MacLane space of type $K(G, n)$ is per definition a space $X$ (homotopy equivalent to a CW-comples) such that $\pi_nX \cong G$ and all other homotopy groups are zero [thus, its very definition uses homotopy groups]. The main use is the following: Homotopy classes $[Y,X]$ for $X$ a $K(G,n)$ are in one-to-one correspondence with elements in the cohomology $H^n(Y;G)$.
For example, $\mathbb{CP}^\infty = BU(1)$ is a $K(\mathbb{Z},2)$. Thus, we can immediately conclude that homotopy classes $[Y,BU(1)]$, i.e. isomorphism classes of line bundles on $Y$ (if $Y$ is paracompact), are in one-to-one correspondence with classes in $H^2(Y;\mathbb{Z})$.

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