Motivation/intuition behind the definition of delta-functors and related concepts

Question

1. Why are $\delta$-functors between abelian categories $\mathfrak{A}$ and $\mathfrak{B}$ defined as a collection of (right) derived functors, and connecting morphisms $\delta^i: T^i(A'') \rightarrow T^{i+1}(A')$ (given an original short exact sequence $0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0$ in $\mathfrak{A}$) $$(T^*, \delta^*) := \{(T^i, \delta^i)\}_{i \geq 0}$$ Particularly, in the context of algebraic geometry, what about this definition is geometric ?
2. What is a geometric interpretation of the universality of $\delta$-functors ?
3. What is a geometric interpretation of effaceable $\delta$-functors, and by extension, theorem 2.2.2 in Grothendieck's Tohoku paper stating that effaceability implies universality ?