On the suggestion of DamienC, I'm converting my comments into an answer. I didn't do this before because I don't really know the answer to the deeper question of why the microlocal sheaf category is the correct thing to give the Fukaya category (and I mostly just say things which are either obvious or wild speculation).
Some naive reasons why it's reasonable to expect local information on the skeleton to know everything:
It's true for cotangent bundles and represents a nice categorical enhancement of many historical results about cotangent bundles, starting with Viterbo/Abbondandolo-Schwarz results that symplectic homology of a cotangent bundle is the homology of the loopspace of the zero section.
Liouville flow retracts a Weinstein manifold onto its skeleton, so everything about symplectic topology of the completion should be determined by the germ of the manifold along the skeleton.
More speculatively:
- In mirror symmetry, you could think of $\mathbf{R}^n$ as a torus with very large radius (like how $\mathbf{R}$ is a circle with very large radius). When taking the dual Lagrangian torus fibration, a non-compact $\mathbf{R}^n$ fibre should therefore be dual to a point (circle with very small radius). You see examples of this in recent work by Lekili and Polishchuk (https://arxiv.org/abs/1705.06023), where they find mirrors to punctured surfaces which are nodal curves: the structure sheaf at the node is mirror to a wrapped Lagrangian brane which is a non-compact $\mathbf{R}$ going off to the puncture. So what is mirror to the Lagrangian $\mathbf{R}^n$-fibration of the cotangent bundle by cotangent fibres? The zero section itself. Quite how you figure out that the derived category should be replaced by the category of microlocal sheaves, I don't know.
In fact, Kontsevich explains the motivation for the conjecture in his own words in his paper "Symplectic geometry of homological algebra": https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf