Why was it reasonable to ask what the higher K-groups are? To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the following question to myself:
I understand that $K$-theory had started with the Grothendieck-Riemann-Roch in mind, and that the only thing that was needed for that purpose from $K$-theory was just to define the Grothendieck Group ($K_0$). Once the idea of the Grothendieck group was established, this was generalized to topological spaces, as well as for other kinds of modules. Then comes the step I don't understand -- it seems that people were then trying to find the higher $K$-groups that would make $K$-theory into a cohomological theory. Milnor came up with Milnor $K$-theory, which I understand from wiki is different from later notions of higher $K$-theory. But why would one leap from the concept of the Grothendieck group to thinking that this construction is the $0^{th}$ step in a cohomological theory? What was the context/motivation for that?
 A: After having defined K0, a natural things to do is to study its functoriality properties.
You do that, and you notice some exact sequences... that you happen to be able to extend a 
bit by using the functors K1, and then later K2.
It is then natural (specially if you know what cohomology is) to try to find long exact sequences that extend the above sequences... A lot of people tried to do that.
Quillen's brilliant idea was to define algebraic K-theory as the homotopy groups of an appropriately constructed space. In that way, the long exact sequences came as natural consequences of known long exact sequences in topology.
 Slogan: Homotopy theory is the mother of all long exact sequences.
Aside: I also recommend reading Thomason's work on algebraic K-theory of schemes: it's beautifully written!
A: 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.)
