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 Thomasson's Thomason's work on algebraic K-theory of schemes: it's beautifully written!
