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.