There are a lot of answers and this is an old question, but I'm surprised nobody mentioned the following perspective. This is admittedly more of a motivation for studying homological localization of spectra rather than spaces. In fact, I find localization of spaces to be somewhat mysterious from this perspective.

The perspective is that the $\infty$-category $Sp$ of spectra is an interesting stable, presentably symmetric monoidal $\infty$-category. Now in 1-categories, there's a school of thought which advocates studying rings via their categories of modules, or more generally studying schemes via their categories of quasicoherent sheaves (which are presentably symmetric monoidal abelian categories), and even by extension viewing presentably symmetric monoidal abelian categories as a sort of generalized space. In the $\infty$-categorical setting, it's similarly natural to view an arbitrary stable, presentably symmetric monoidal $\infty$-category as a sort of generalized space, in the spirit of derived algebraic geometry. Of course, $Sp$ is the category of modules of the sphere spectrum, so it's not really necessary to think about non-affine derived schemes or general symmetric monoidal categories to think from this perspective -- we're just studying the derived $Spec$ of the sphere spectrum $\mathbb S$.

Anyway, this means that the 1-categorical analog of thinking about homological localizations is thinking about the supports of modules / quasicoherent sheaves, and considering localizing a scheme / ring away from this support. To be honest, I don't know enough algebraic geometry or commutative algebra to really say why this is important, but it certainly sounds like a reasonable thing to be interested in, which you could imagine motivating from internal geometric / algebraic considerations without ever dreaming up the concept of arbitrary localizations.

I guess the 1-categorical analog of your question is: to what degree is localizing away from the support of a module/sheaf especially fundmental among all possible Serre quotients of a category of modules / sheaves? Or one could equally ask the question in the setting of Verdier quotients of triangulated categories. Again, I don't really know enough to answer this, but I'm pretty sure there are people who could.

Tentatively, I'd say that what localizing away from the supports of modules does is to give you a handle on (certain) Zariski-open subschemes of your scheme. I might even venture that it's not quite so fundamental as it seems classically insofar as the Zariski topology is not the most important topology -- it's just the class of open map into our schemes which is easiest to access with older technology. Perhaps one should really be fundamentally interested in the etale or motivic version of homological localization, whatever that is.

There are a lot of answers and this is an old question, but I'm surprised nobody mentioned the following perspective.

Question 1: Why localize spaces with respect to homology?

Answer 1: Don't!

(Except as a technical tool.)

Instead, take an interest in homological localizations of spectra.

I think that this answer aligns with the practice of "most" homotopy theorists -- with notable exceptions! (Some interesting results of Neisendorfer come to mind.) The sense that localizing spaces with respect to homology is a "violent" and borderline "unnatural" operation, which perhaps should be replaced by something else, was one motivation for this question of mine. I think I would say (without being terribly knowledgable in the subject) that for unstable localizations, there's less reason to be more interested in the homological case than in the general case.

So we're left with a different, related question:

Question 2: Why localize spectra with respect to homology?

This one I think admits a more systematic answer, which first requires answering a more basic question:

Question 0: Why study spectra at all?

Of course, there are many good answers to this question, but let's look at one:

Answer 0: Brave new algebra + Derived algebraic geometry!

I think these ideas have become commonplace today, but let's sum them up:

• Waldhausen's philosophy of "brave new algebra" stipulates that we study the sphere spectrum $\mathbb S$, spectra, ring spectra, modules over ring spectra, etc. because they form a world of algebra which is better-behaved, more structured, and more fundamental than their decategorifications living in the "cowardly old" world of the ring $\mathbb Z$, discrete abelian groups, discrete rings, discrete modules, etc.

In particular, the most fundamental aspects of commutative algebra are decategorifications of analogous, even more fundamental aspects of brave new algebra.

• One of the most fundamental aspects of commutative algebra is that it constitutes the local / affine part of the whole subject of algebraic geometry. Categorifying, brave new algebra constitutes the local / affine part of the subject of derived (or spectral) algebraic geometry.

So from this perspective, we can refine our question:

• The category $Sp$ of spectra is analogous to the category of abelian groups: we have that $Sp = Mod(\mathbb S)$ is the category of modules over the initial $E_\infty$-ring spectrum $\mathbb S$ just as $Ab = Mod(\mathbb Z)$ is the category of modules over the initial ring $\mathbb Z$. Equivalently, $Sp = QCoh(Spec(\mathbb S))$ is the category of quasicoherent sheaves over the terminal spectral scheme $Spec(\mathbb S)$ just as $Ab = QCoh(Spec \mathbb Z)$ is the category of quasicoherent sheaves over the terminal scheme $Spec(\mathbb Z)$.

• If $E \in Sp$ (or more generally $E \in Mod(A)$ for an $E_\infty$ ring spectrum $A$, or $E \in QCoh(X)$ for a spectral scheme $X$), then localizing with respect to $E$-homology means passing to quasicoherent sheaves on the open subscheme $Supp(E) \subseteq X$ where the quasicoherent sheaf $E$ is supported.

Question 2 (bis): Given a category of quasicoherent sheaves $QCoh(X)$, why study it via the supports $Supp(E) \subseteq X$ of its objects $E \in QCoh(X)$?

And here I think we have a question which we can actually answer, in a few ways:

Answer 2:

1. Geometrically, what we are doing is getting a handle on the Zariski-open subschemes of our scheme $X$. This is a fundamental thing to do -- it's hard to imagine doing anything if you can't understand the Zariski topology of $X$.

2. The real question becomes: when studying the Zariski topology of $X$, why should we immediately reach for the categorical description in terms of localizations of $QCoh(X)$? After all, in the non-derived world, we tend to get a handle on the Zariski topology of a scheme much more directly.

In some sense, the answer is that this is a "historical contingency": 30-40 years ago the machinery of derived algebraic geometry was not in place, but the categorical data of $QCoh(X)$ was something people could get their hands on, so they worked with that. It's much like the situation in noncommutative geometry: when it's unclear what the definition of a "noncommutative scheme" $X$ should be, but at least clear what $QCoh(X)$ should be in certain cases, you just do what you can with the category $QCoh(X)$ and you make progress.

3. One might argue that the Zariski topology is not the most important topology (e.g. perhaps the etale or Nisnevich topologies are more important), and consequently we shouldn't put so much emphasis on localizations homological or otherwise. From this perspective, it's again a historical contingency that the Zariski topology was the easiest to get at with older technology (just as it was in the underived world.)

4. You could (and some do!) turn this situation on its head and argue that the functorial viewpoint on algebraic geometry, where one studies $X$ via $QCoh(X)$, really is more fundamental after all. Then we return to the question: why are homological localizations particularly interesting among all localizations? For this I'd appeal to the fact that they have more structure; e.g. the structure of a recollement.

Finally, I think it goes without saying that the case where $X = Spec(\mathbb S)$ is the terminal scheme is particularly fundamental -- just as studying the open subschemes of $Spec(\mathbb Z)$ (i.e. primes!) is so fundamental to algebraic geometry that it goes without saying.