Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ has to be representable. The latter means that if $X,Y$ are schemes, $h_X,h_Y$ their Yoneda representations and we have natural maps $h_X \to \mathcal{X}, h_Y \to \mathcal{X}$ then the fiber product $h_X \times_{\mathcal{X}} h_Y$ has to be representable in "usual" sense by a scheme. That's a clear formulated technical condition that one can start to verify.

Question: What is the intuition/philosophy one should have in mind associating with this condition on representability of the diagonal $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$? What makes this condition so "interesting" in order to study stacks? What is the wittness of this condition?

Recall then we talking about schemes a scheme $S$ is called separated if the diagonal map $S→S×S$ is a closed immersion. It allows to "transfer" in certain way a "category theoretical Hausdorff axiom" the algebraic geometry, since the classical one fails for Zariski topology.

For schemes the property "be separated" is is often used in following sence: Let $f: X\to Y, g:Y \to Z$ morphisms, composition $g \circ f$ has certain property P well behaving under pullbacks and $g$ separated. Then $f$ has also P.

Is the the motivation for representability of $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ of similar manner?

That is my question is which intuitive property is "transfered" by imposing the representability for the diagonal map to the world of stack?