If I understand correctly, homotopy groups behave well under products, and so it's reasonable that fiber bundles give long exact sequences.
What is weird to me is why there is even an analog of excision for higher homotopy groups. For homology it makes sense that $H(X,A) \cong H(X/A)$, because heuristically we imagine refining everything so simplexes lie in a small nbr of $A$ or a nbr of a complement. However, while I follow the proof of the homotopy excision theorem, the resulting $\pi_n(X,A) \cong \pi_n(X/A)$ theorem is weird; it has a bunch of conditions on $A, (X,A)$ being very simply connected for it to work, which hints it is somewhat unnatural.
Is there a better frame-work to view the homotopy excision?