I'm going to give an algebraist's perspective. First let's discuss homological algebra (which has roots in topology). There's a quote (attributed, I think, to Connes) that a great mystery of homological algebra is the power afforded by the formula $d^2=0.$ This formula is indeed mysterious and the resulting study of chain complexes yields powerful results. But an even more powerful formula, that people rarely think about in this way, is 
$$a+d = b+c \implies (a,b)\sim (c,d).$$ 
This fundamentally powerful equivalence relation on pairs of natural (or real positive) numbers yields wonderfully versatile number systems, which have much better properties than the original semigroups. But mathematicians don't think about $\mathbb{Z}$ or $\mathbb{R}$ as equivalence relations on pairs: rather, we think of them as fundamental objects in their own right: the equivalence relation is simply a way to access them. The same can be said about differential graded categories: the whole formalism of projective resolutions and quasiisomorphisms is just a way to access objects like the Serre dualizing sheaf that are fundamental, and that the more rigid formalism of abelian categories is insufficient for. In this sense, the category of complexes is just one particularly elegant choice for "softening" the abelian category (this is a term I first heard from Kontsevich): others exist as well. (See e.g. You Qi's [hopfological algebra][1].)

The modern interest in topology, as I see it, comes from the fact that it turns out that some *fundamental* objects interesting to all mathematicians, including algebraists (e.g. K-theory spectra, tangent complexes in characteristic p, certain cyclic and motivic constructions) are not accessible through pure algebra, but can be "evaluated" on topological spaces, and can then be reconstructed from these evaluations using techniques related to modern homotopy theory. It is somewhat miraculous that so many constructions interesting to algebraists have good formulations in terms of something topologists had thought about for a long time, but this is just one of those miracles we have come to expect from any elegant mathematical theory.

The important questions of homotopy theory from this point of view are of the form "what are the right definitions, when do they work, and how do we compute with them". For example, suppose $A$ is some algebraic gadget (algebra, Lie algebra, etc.) Consider the group of automorphisms of structure, $Aut(A)$. In really nice cases, this is a Lie group, it's locally determined by its Lie algebra (of derivations), and it is the stabilizer of a point in a well-behaved classifying orbifold. It turns out that if one allows to supplement geometric ideas by topological ones, the class of "really nice cases" expands dramatically. But one needs to understand the appropriate meaning of the words "Lie group", "local", "Lie algebra", etc., and what one can do with them. This is done in very strong generality in Jacob Lurie's work on [Moduli spaces][2].


  [1]: http://arxiv.org/abs/1205.1814
  [2]: http://www.math.harvard.edu/~lurie/papers/moduli.pdf