  The dramatic slowing down of relaxation dynamics of liquids approaching the
glass transition remains a highly debated problem, where the crux of the puzzle
resides in the elusive increase of the activation barrier $\Delta E(T)$ with
decreasing temperature $T$. A class of theoretical frameworks -- known as
elastic models -- attribute this temperature dependence to the variations of
the liquid's macroscopic elasticity, quantified by the high-frequency shear
modulus $G_\infty(T)$. While elastic models find some support in a number of
experimental studies, these models do not take into account the spatial
structures, length scales, and heterogeneity associated with structural
relaxation in supercooled liquids. Here, we propose that viscous slowing down
is controlled by a mesoscopic elastic stiffness $\kappa(T)$, defined as the
characteristic stiffness of response fields to local dipole forces in the
liquid's underlying inherent states. First, we show that $\kappa(T)$ -- which
is intimately related to the energy and length scales characterizing
quasilocalized, nonphononic excitations in glasses -- increases more strongly
with decreasing $T$ than the macroscopic inherent state shear modulus $G(T)$ in
several computer liquids. Second, we show that the simple relation $\Delta
E(T)\propto\kappa(T)$ holds remarkably well for some computer liquids, implying
a direct connection between the liquid's underlying mesoscopic elasticity and
enthalpic energy barriers. On the other hand, we show that for other computer
liquids, the above relation fails. Finally, we provide strong evidence that
what distinguishes computer liquids in which the $\Delta E(T) \propto
\kappa(T)$ relation holds, from those in which it does not, is that the latter
feature highly granular potential energy landscapes, where many sub-basins
separated by low activation barriers exist. [Rest of abstract abridged]
