How does a stator convert velocity into pressure? I'm looking for, hopefully, a qualitative answer to how a compressor increases pressure. I understand the equations that predict this result but I can't picture it in my head.
A rotor spins, deflecting air into the engine and increasing the air's velocity. Next, there are stationary fins that this relatively high speed flow collides with. Somehow this contact takes away velocity and adds pressure. This crucial step is the part I don't understand.
 A: A stator vane is nothing but a little wing, so it is probably best to first explain how a wing works. Please follow the links over to Aviation SE for a detailed explanation. Here is the gist of it:
A wing creates lift by accelerating the flow of air downwards. This is called downwash and is also the effect which creates the slipstream behind a spinning propeller. The stator vane is not spinning itself, but is placed into a rotating flow, such that the flow around it is very similar to that of a wing or a propeller blade.
The rotational flow component is transformed into an axial acceleration which in turn is converted into ram pressure by the narrowing flow path in a compressor. If you look at it at an energy level, kinetic (rotational) energy is converted into potential (pressure) energy.
A: It's correct the stator's only purpose is to redirect flow to the next set of turbine blades by redirecting the downward flow from the turbine blades, back to the top of the next turbine.This is what as seen as and S-flow. Here is a quick visual of what it looks like.
refer here for more information. 
