Recently, Akhil Mathew has published papers on $K(1)$-local theory: On $K(1)$-local $\mathrm{TR}$ and Remarks on $K(1)$-local $K$-theory.
What is the motivation of $K(1)$-local theory? What does $K(1)$-local theory correspond to in the area of algebraic geometry?
(I'm very surprised that the following hasn't been mentioned in the comments so far -- I thought it was conventional wisdom!)
The main reason to study $K(1)$-local homotopy theory is that it is the first step in the infinite ladder of chromatic homotopy theory. The idea of this subject, in brief, is that the category of spectra is "controlled" by the chromatic heights, of which there is one for every pair $(p,n)$ where $p$ is a prime and $n \in \mathbb N \cup \{\infty\}$ (but where $(p,0)$ is identified with $(p',0)$). One formalization of this philosophy is the thick subcategory theorem, a deep theorem about the structure of the category of finite spectra which goes along exactly these lines. The chromatic heights are the analog in stable homotopy theory of the primes themselves in ordinary algebra.
The 0th rung on this ladder (i.e. height 0) is rational homotopy theory, related to localization at $H\mathbb Q$. The first rung (i.e. height $(p,1)$) is $K(1)$-local homotopy theory (which depends on a prime $p$, omitted from the notation), related to localization at $KU/p$. The higher chromatic heights quickly become very mysterious, but height 2 is closely related to elliptic cohomology and toplogical modular forms.
So my answer would be: we would like to understand stable homotopy theory. The zeroth step is to understand rational homotopy theory. Here, the state of knowledge is pretty good. The first step is then to understand $K(1)$-local homotopy theory.
The work of Akhil Mathew et al. really is a continuation of Thomason's: the goal is to consider algebraic $K$-theory and its sibblings (such as $TC$) and to see them as cohomology theories of (affine) schemes. A natural question is then to understand descent properties (i.e. local to global principles): $K$-theory governs intersection theory through Chern classes (and Atiyah–Hirzebruch–like spectral sequences produce filtrations whose associated graded are higher Chow groups), and if we consider intersection theory in cohomologies for which étale descent holds (e.g. $\ell$-adic cohomology), it is natural to wonder what are the structural properties of $K$-theory étale locally. The fact that $K(1)$-localization of algebraic $K$-theory almost gives its étale sheafification (as observed by Thomason a long time ago) and the fact étale cohomology in fact satisfies $h$-decent (essentially because of proper base change formulas, as observed by Grothendieck in SGA 4), makes that it is natural to study descent for topologies related to (or generalizing the) $h$-topology, at least after $K(1)$-localization. The point, for applications of such results, is that, locally for the $h$-topology, singularities have great tendency to disappear (for quasi-excellent schemes, the theorem of De Jong–Gabber literally says that), and the local descriptions are essentially determined by valuation rings which are huge but with nice structural properties. Furthermore, topologies such as the $h$-topology are not subcanonical, so that non-trivial new identifications follow from this kind of descent results (we can also study Milnor excision, relation with homotopy invariance…). There are furthermore fundamental results such as the Quillen–Lichtenbaum conjecture (consequence of Bloch–Kato) which explain how to deduce structural properties of $K$-theory from its étale variant.
Of course, we can decide not to ignore that $K(1)$ is just one step in the scale of complexity of chromatic homotopy theory. Working $K(0)$-locally simply means that we work with $\mathbb{Q}$-coefficients, and indeed, algebraic $K$-theory with $\mathbb{Q}$-coefficients satisfies étale descent (and Weibel's homotopy invariant $K$-theory with $\mathbb{Q}$-coefficients satisfies $h$-descent), and that fits in the $K(1)$-local picture. The perspective of higher chromatic localizations of algebraic $K$-theory is studied by Markus Land, Akhil Mathew, Lennart Meier and Georg Tamme in Purity in chromatically localized algebraic $K$-theory.
