I think a key point is that algebraic K-theory is defined not only for rings, but also for schemes (and other kinds of "generalized spaces" in algebraic geometry). If you believe that generalized (Eilenberg-Steenrod) cohomology theories are useful/interesting in algebraic topology, then it is also reasonable to think that they might be interesting in algebraic geometry, and algebraic K-theory is in some sense the simplest and most widely studied such theory, although yes, computations are very hard.

Some other motivation:

Algebraic K-theory allows you to talk about characteristic classes of vector bundles on schemes, with values in various cohomology theories, see for example Gillet: K-theory and algebraic geometry.

Algebraic K-theory is intimately connected with motivic cohomology and algebraic cycles, see for example Friedlander's ICTP lectures available on his webpage, especially the 5th lecture on Beilinson's vision: http://www.math.northwestern.edu/~eric/lectures/ictp/

One of the major themes in arithmetic geometry is the study of special values of motivic L-functions. These values capture a lot of deep arithmetic invariants of number fields and varieties over number fields, and they seem to be mysteriously related to many other things, for example orders of stable homotopy groups of spheres. There are many results and conjectures about these values, most famously the Clay Millennium Birch-Swinnerton-Dyer conjecture, and in many versions of these conjectures, algebraic K-theory plays a crucial role. See for example the survey by Bruno Kahn in the K-theory handbook, also availably at his webpage: http://people.math.jussieu.fr/~kahn/preprints/kcag.pdf

There are also many other useful things in the K-theory handbook, such as the lectures by Gillet on K-theory and intersection theory, also available here: http://www.math.uic.edu/~henri/preprints/K-Theory_Chow_Groups-6.pdf