Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book on algebraic K-theory (I am not an expert) I see various complicated very ingenious constructions which become equivalent for mysterious (to me) reasons. What I do not understand, what problem exactly Quillen tried to solve?
Are there any properties that any higher algebraic K-theory is expected to satisfy a priori
like long exact sequences (assuming that $K^0$ is known)? I realize that in any formulation it has to be functorial, but this does not tell much unless more information is known.
In order to give an example of an answer which would be kind of convincing for me, let me try to summarize my understanding of topological $K$-theory. Since I am not an expert, this understanding is incomplete and down to earth, but this is what I am looking for in the algebraic case. In what follows, the first two paragraphs are about the motivation of topological $K$-theory, while the other two are about concrete applications of it.
1) Grothendieck defined algebraic $K^0$ ring for schemes in order to formulate his generalization of Riemann-Roch-Hirzebruch theorem. The construction used algebraic vector bundles. Topological $K^0$
was defined by analogy using topological vector bundles.
2) Higher topological $K$-groups were introduced in order to have the Mayer-Vietoris long exact sequence for pairs of spaces.
3) There is an important construction of elements of $K^0$-groups: any elliptic (pseudo-) differential operator defines an element in $K$-theory of the tangent bundle. This construction is necessary for the formulation of the Atiyah-Singer theorem.
4) After some identifications based on the Bott periodicity, higher rational $K$-ring becomes isomorphic to the rational cohomology ring via the Chern character. Thus $K^*/torsion$
is a lattice in $H^*(\cdot,\mathbb{Q})$ which is different from the lattice $H^*(\cdot,\mathbb{Z})/torsion$. This canonical and non-obvious integral structure was used to prove some non-embeddability theorems.
Best Answer
I am neither a K-theorist nor a historian, so I don't know all the things the led to Quillen to his definition(s) of higher K-theory, but I thought I'd mention one striking application that can be found in his original paper. The Chow group of a variety $CH^p(X)$ is the group of codimension $p$-cycles modulo rational equivalence. For lots of reasons, it was desirable to express this in terms of sheaf cohomology. For $p=1$, $CH^1(X)=Pic(X)= H^1(X, O_X^*)$ was known for a long time. This can be recast into K-theoretic terms, by observing, using Bass' definition, that $O_X^*$ can be identified the sheaf associated to $U\mapsto K_1(O(U))$. I believe Bloch extended this to $CH^2$ using Milnor's $K_2$. And finally Quillen proved that $CH^p(X)=H^p(X,K_p(O_X))$ for any regular variety, and any $p$ using his definition.