Path integral derivation of extended TQFT
The original motivation for extended TQFTs (as introduced by Freed, Lawrence, Baez-Dolan) is indeed giving a finer form of locality, as explained by Dmitri Pavlov. However I think there are two quicker, and arguably more physical, ways to see n-categorical structure in n-dimensional QFTs.
The first is not really about the states of a QFT (as axiomatized in the Atiyah-Segal formalism) but their algebras of observables (extending the distinction between geometric and deformation quantization in the context of quantum mechanics). Namely the theory of factorization algebras as developed in the book of Costello-Gwilliam extracts from the same data as the path integral an n-dimensional factorization algebra of observables. In the topological context such a factorization algebra is the same as an $E_n$ algebra, which is the same as a very connected $(\infty,n)$-category (one with one object, one 1-morphism, ...all the way down).
The second comes from thinking of what ELSE there is in a QFT beyond the path integral -- the most important being the structure of defects of various dimensions. Of these the richest is the notion of a boundary theory (or "boundary condition") for a QFT, which is very loosely "things we can put on the boundary and couple to our theory" -- something like a QFT of one dimension lower that lives on the boundary of manifolds where the bulk carries our given QFT.
In any case, boundary theories in an $n$-dimensional QFT naturally form something like an $(n-1)$-category, which in the cobordism hypothesis formalism for extended TQFT is closely related to what you'd attach to a point. Namely, as morphisms between any two boundary theories you can consider codimension 2 defects that are interfaces between the two theories (think of dividing the boundary of a half-space in $R^3$ into upper and lower halves with a 1-dim interface on the intersection). As 2-morphisms you can consider interfaces between interfaces, and so on and so forth.
To me this is the most compelling way to see that higher categorical structure is physically natural/meaningful. A (somewhat criminal) paraphrase of the cobordism hypothesis says that a fully extended TQFT is determined by its collection of boundary conditions. [Really boundary theories are morphisms between the unit and the object attached by the TQFT to a point, which in general needn't determine this object, but it's a decent ansatz.]
The physics motivation for extended QFTs (and not just TQFTs) comes from the locality principle (no spooky action at a distance).
The mathematical expression of locality is the descent property for extended QFTs. See, for instance, Higher Algebraic Structures and Quantization by Daniel S. Freed.
Specifically, the assignment to X of the symmetric monoidal (∞,n)-category of extended QFTs with bordisms equipped with a map to X is a stack of symmetric monoidal (∞,n)-categories with respect to X.
This claim fails unless we work with QFTs extended all the way down to points, since proving the descent property requires cutting bordisms all the way down to points.
An informal path integral construction produces an extended QFT starting from another functorial field theory (quantum or classical) on Y and performing pushforward along a map Y→X. The pushforward simply integrates over spaces of n-dimensional manifolds mapping to Y that have the same image in X.