Why is shaft work equal to $VdP$? Mathematically it makes sense to me, because there is -PdV work occurring plus you must account for the change in flow work [d(PV)], so -PdV + PdV + VdP = VdP.  However, I'm having a difficult time conceptualizing how expansion work and a change in flow work both create shaft work.
 A: Why is shaft work equal to VdP?
It isn't. 
@Chet Miller made me realize that I didn't fully understand your question. Flow work is input to system and is part of the energy that produces shaft work (output). It is not the shaft work itself. It is included as part of the enthalpy entering and exiting a control volume. An example is the input for a turbine that produces shaft work. The following is by way of explanation.
There are basically two forms of thermodynamic work- Boundary work ($PdV$) and flow work ($VdP$).
Boundary work primarily applies to closed thermodynamic systems, that is, systems that do no involve mass transfer with the surroundings. It refers to the expanding or contracting of the system boundaries. The common example used is the expansion or compression of a gas in a cylinder fitted with a piston. 
Flow work applies to open systems where there is flow into and out of what is referred to as a "control volume". As @Chet Miller pointed out it is the work done to push fluid into and out of the control volume where a pressure differential exists at the inlet and outlet of the control volume.
The first diagram below shows a generic control volume that accounts for various possible heat and work transfers to or from the control volume, as well as changes in the kinetic and or potential energy of the working fluid into and out of the volume. The flow work is accounted for in the PV term of the entering and exiting enthalpy of the fluid.
The second diagram shows a special case of the first where certain terms are removed  (changes in kinetic and potential energy, and heat transfer) as applied to adiabatic turbines (other than gas turbines), compressors or pumps. For the case of a steam turbine, the turbine work output is simply the change in enthalpy of the steam at the inlet and outlet of the turbine. That enthalpy includes the $VdP$ work needed to push the steam into and out of the turbine.
Hope this helps.


