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.
This is addressed to Paul Siegel's answer, which I find misleading, and
not just because it is not true that maps that induce the same
map on homotopy groups are homotopic. (The famous and open Freyd conjecture
asserts that maps of finite spectra that induce the same map on homotopy
groups are homotopic, but that is digressive).
It is the comments about stable homotopy theory that I find most misleading.
Good modern categories of spectra are symmetric monoidal under the smash
product, and in such categories $E_{\infty}$ ring spectra as first defined
in 1972 are equivalent to commutative monoids (as defined in any symmetric
monoidal category). The higher homotopies are then hidden in the definition
of the smash product. For purposes of calculation, this is extremely
convenient and useful, and there is little point to invoking higher
category theory of any sort. For example, people doing serious computations with TMF
don't invoke higher category theory.
Moreover, higher category theory should not be confused with just the theory
of $(\infty,1)$-categories. Those are central to categorical homotopy
theory, but higher category theory has many other incarnations. For
example, $2$-category theory as developed in Australia is hugely rich and
powerful in applications. In the hands of Riehl and Verity it has very
important applications even to the theory of $(\infty,1)$ categories, but
it is very concretely useful in a slew of other areas, including stable
homotopy theory. Moreover, there are quite
distinct and quite different theories of $n$-categories, which start with
sets and categories at levels $0$ and $1$, and $(\infty,n)$-categories,
which can start with simplicial sets and simplicially enriched categories
at levels $0$ and $1$ (there are other possible choices).
Both 2-category theory and $(\infty,1)$-category are examples where there are lots of applications outside the theory itself, just as is true of ordinary category theory.
In addition to the sources others have cited, there is a book ``Towards higher
categories'' with articles that discuss various aspects of higher
category theory and that may give some idea of the why as well as the wherefore.
Best Answer
The idea of considering higher K-groups comes from topology, and is due to Atiyah, Bott, and Hirzebruch. Atiyah and Hirzebruch defined topological K theory and observed that Bott periodicity says that $K(X)$ is more or less the same as $K(S^2X)$. This suggested to them defining a generalized cohomology theory of period 2 by using all the groups $K(S^nX)$ (this was the first example of a generalized cohomology theory). Once one realizes that topological $K^0$ can be extended to topological $K^n$, it does not take much imagination to suggest that algebraic $K^0$ also has an extension to algebraic $K^n$. (Of course, finding this extension was much harder than guessing it existed.)