We study small disturbances of the plane periodic Couette flow in the 3D Navier-Stokes equations in the limit Re -> infinity. Specifically, we are interested in obtaining a mathematically rigorous understanding of the subcritical instability. For sufficiently smooth perturbations we prove that the subcritical transition threshold as ~Re^(-1) and prove that all possible instabilities near this threshold are driven only by the secondary instability of "streak" solutions driven by the lift-up effect. For rougher data we estimate the threshold as being at least Re^{-3/2}. In all these regimes, the fast mixing of the solution due to the mean shear dominates the dynamics for long times, driving a rapid homogenization via mixing-enhanced dissipation effects and inviscid damping -- hydrodynamic analogue of Landau damping in plasma physics.