With the comments having clarified the question a bit, let me just say that the a priori motivation for these particular axioms was just whatever was going through Grothendieck’s and Heller’s heads when they wanted to build something that looks like a 2-functor of categories of diagrams and Kan extensions between them. Certainly (Der3),(Der4) (the mere existence and pointwise nature of the Kan extensions) are unavoidable for this, while (Der1) (preservation of coproducts) might fall out of considering the very little extent to which a derivator might be like a stack. (Der2) (conservatively of the underlying diagram functors) is less obviously unavoidable but certainly makes you feel like you’re looking at diagrams, and is one way of guaranteeing that every diagram in a derivator can be built out of constant diagrams, so that a derivator is really like a refinement of its base.

(Der5) (weak quotientness of the underlying diagram functor for diagrams with free domain) is probably the least obvious, but is well motivated by the usefulness of distinguished weak colimits in triangulated category theory, and some parts of unstable homotopy theory. It is also not necessary for the strongest a priori justification of the other axioms: Cisinski’s theorem that the derivator of spaces is freely generated by the point. In particular, note that this theorem fails without (Der1) and (Der2), and is senseless without 3 and 4.

(Der5) is necessary to consider which derivators are representable by $\infty$-categories or model categories, since such derivators certainly satisfy (Der5). One goal of my thesis was to study the extent to which it’s sufficient. The answer is "not quite", but a similar axiom saying that a derivator respects all lax pushouts, rather than merely the lax pushout $0\to 1,$ is a sufficient replacement. The only other missing ingredients are preservation of certain *coinverters*, or *localizations*, a type of 2-colimit whose preservation fits in nicely with (Der1) but was not originally noticed, as well as various technical details about size.