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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.
The main principle is the conversion of dynamic pressure to static pressure.
The pathway of the fluid is such that initially velocity decreases due to the mass flow rate equation (I'm not writing equations as you mentioned you knew them). The important idea here has to do with the energy lost in the ducts as it converts kinetic energy into internal energy and therein lies the pressure increase. Think of a stressor matrix in 3D with the sum of the principle components, which we know to be equal to hydrostatic pressure
As the fluid enters it first hits the compressor which increases the pressure by doing work (adding energy) to the flow. This pressure is dynamic pressure as the velocity increases at the compressor stage ( $q = 0.5 * \rho * V^2$).
The fluid then enters the diffuser stage where the stators turn this dynamic pressure into static pressure by slowing down the flow.
So to summarize, the compressor adds velocity to the flow by doing work (adding energy) to the flow. This energy is then converted from dynamic pressure to static pressure by the diffuser as it slows down the fluid.
In terms of a "qualitative" answer, I think the best way to think about a compressor is in kneading or shaping dough (like in baking). Imagine you have long strip of bread dough on a working surface. Using your hands as paddles, if you were to take turns pushing the dough, alternating with each hand, the dough would turn into a smaller and smaller strip with each pass. This is similar to rotors and stators ("paddles") compressing air in a compressor.
In a compressor with multiple stages, you have many stages of rotors/stators that gradually increase the static pressure, but the way in which it increases the pressure is dynamic. With a rotor, you are imparting momentum (and with it velocity) to the air. Next, when the air encounters the stator, the momentum and veclocity "crash" into the stator. Because the air is a compressible fluid, this action increases the density. An increase in density is the definition of compression.
The actual mechanics are a little more elegant than "crashing" (think of paddling the dough), but I think you get the picture. Stators are designed to change the direction of the flow, and this causes a "build up" of air as the air switches direction. In addition, the exit area (or "throat") of the stator further restricts free flow and causes a pressure to build.
Hope that helps!
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.'
So the stator's purpose is to redirect the flow into straight lines parallel to the center shaft. The stators are not responsible for pressure increases in that sense but still do have some responsibility. But the narrowing compressor is the main reason of the pressure increase. The stators just help to force the flow uniformly into the smaller space. Although, the stators still do have some responsibility in increasing pressure: since air is compressible, when the fluid's particles hit onto the stators, density increases and therefore compression does as well.
The stator's 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 an S-flow.
