TITLE: Quillen's motivation of higher algebraic K-theory
QUESTION [25 upvotes]: 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.

REPLY [7 votes]: Thanks to the link  http://www-irma.u-strasbg.fr/~loday/DanQuillen-par-JLL.pdf mentioned above by Philippe Gaucher, I learned something about Quillen's motivation of the $+$-construction (though still there are many gaps in my understanding of further developments). Below is a very short summary of what I could understand. The original reference is 
"On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field",
Daniel Quillen,
Ann. Math., Vol. 96, No. 3 (Nov., 1972), pp. 552-586.
The most surprising thing I learned is that Quillen was solving a problem completely unrelated to algebraic $K$-theory: he computed the group cohomology $H^*(GL_n(\mathbb{F}_q),\mathbb{F}_l)$ of the linear group over finite field with coefficients in another finite field $\mathbb{F}_l$ where $l$ is a prime number not dividing $q$. As a first step based on the Brauder's theory of characters of finite groups, Quillen partly reduced this problem to the topological question of describing the homotopy fiber of the map $BU\overset{\Psi^q-Id}{\to} BU$ where $BU$ is the classifying space of the infinite unitary group, and $\Psi^q$ is the $q$-th Adams operation. He proved that this homotopy fiber has the same homology as the classifying space $BGL(\mathbb{F}_q)$, but the $\pi_1$ group of the former is equal to the abelinization of the $\pi_1$ of the latter. Then Quillen realized that one can attach 2- and 3-cells to $BGL(\mathbb{F}_q)$ in a tricky way to get the space with the desired properties.
Trying to understand and generalize his construction, Quillen observed that for any unital ring $R$ if one attaches 2- and 3-cells in the same way to $BGL(R)$, one obtains space with the same homology, but with the abelinized fundamental group; the space is denoted $BGL(R)^+$.
It immediately follows from the construction that $\pi_1(BGL(R)^+)=K_1(R)$. Quillen was able to prove that $\pi_2(BGL(R)^+)=K_2(R)$; the $K_2$ group was already known at the time. It looks likely that these two computations combined with the general aesthetic beauty and elegance of the $+$-construction were suggestive to study higher homotopy groups of the $+$-construction. However at this point my understanding of the intuition stops, since apparently instead of following this way, Quillen introduced a very different (but eventually equivalent) mysterious Q-construction which was used to define higher $K$-groups of schemes and to prove a Mayer-Vietoris type sequence for them.