The high speed expression, proportional to $v^2$ is the ram pressure, which is wholly a momentum transfer effect and has nothing to do with viscosity - in contrast with the low flow speed Stokes law you cite above.
To understand the ram pressure, which arises particularly for supersonic objects, witness the object is just shoving fluid out of its way, and the latter flows off at some high angle to the trajectory. Think of a stationary object with a flat leading surface with a high speed flow around it. Fluid striking the flat surface gets deflected almost at right angles to the incoming flow. If you tally up the impulse per unit time that the object must be exerting on this flow to effect the change in fluid momentum, it is proportional to the flow rate (which, in turn, is proportional to the flow speed), and also proportional to the individual fluid particle momentum - also proportional to the flow speed. So the product of these two is proportional to $v^2$.
Ram pressure is important in the dynamics of stars, galaxies and so forth as these cosmological entities run through interstellar gas and dust.
Edit: see ja72's comment: "Both are in effect, but one dominates over the other at different speeds". I always kind of guessed that - $i.e$ that one might use a model of the form $F = \alpha v + \beta v^2$ but I've never been too sure - because the flow shape arising from the ram pressure changes (delfexion angle changes) with speed, so one has other variables in the momentum tally other than what I've described. Therefore, I guess that a more complicated model other than $F = \alpha v + \beta v^2$ holds at in-between speeds. But you would have to ask a fluid dynamics expert about this. Maybe you should ask the question as well, "what model holds at in-between speeds"?