Being a postulate of AWT, Aether concept cannot explain by itself, what the Aether is or why our Universe exists at all - but it can explain, why it appears by the way, we are observing it. By AWT the gradient driven reality appears as being formed by fractal nested density gradients (i.e. by "fluctuations of fluctuations") of hypothetical dense environment, similar to fluctuations of dense gas. The more fundamental level is, the more the shape of gradients converges to spherical particle aggregates of spherical symmetry, which can be approximated by hypersphere geometry of highest symmetry degree possible.
Because the information and causal energy spreading is mediated by transversal waves exclusively, the effectiveness of energy spreading depends on the volume/surface ratio (as expressed by packing density) of that geometry, if we approximate vacuum as a dense system of closely packed hyperspherical particles (i.e. "kissing hyperspheres"). This ratio exhibits a local supremum for just 3D hypershere geometry, which means, just the space-time formed by gradients of 3D packed spheres can form the most effective environment with respect to energy propagation at distance, which can be expressed by dimensionality dependence of its Lagrangian by principle of least action.
For more than eight dimensions the hyperspheres are contained outside the hypercube with polytope vertices at the centers of the other spheres, which effectively means, such arrangement becomes more sparse and as such disadvantageous with respect to most effective surface energy spreading. Because of tetrahedral conformation of tightest 3D hypershere packing (as given by Kepler's conjecture, i.e. Hilbert's Problem 18, which wasn't solved yet formally), the simple hypersphere model explains both the right angled character of our 3D space, both the upper number of dimensions of observable Universe. The nested kissing 8D hypersphere geometry is closely related to geometry of root vector system of exceptional Lie group E8, on which the Lisi Garrett's "Exceptionally simple theory" is based (see this applet) and the sacred geometry of five elements theory.
The above explanation of vacuum dimensionality is still a somewhat implicit, as it considers right angled geometry of Hilbert hyperspace on background. As an experimental proof could serve the determination of packing density preferred inside of dense particle systems (like the supercritical fluids or plasma crystals), inside which the preferable formation of just 3D clusters should be detected.