Energy meter design/mcp390x
What I am not understanding here is where the voltage part of the watt
  calculation is happening.


Power delivered to a load is voltage x current. That produces watts. If you have a 9V battery and connected to a load and a simple ammeter measured 50mA and you knew the voltage was exactly 9V then the power would be 450mW.

With AC there is an added complication. Power is still volts x amps but it isn't \$V_{RMS}\times I_{RMS}\$ because the voltage waveform and the current waveform may not be precisely in phase. This out-of-phase-ness can be due to the load being a little bit inductive or capacitive. It's a big deal on AC motors because the phase angle between voltage and current may be 45 degrees so the true statement of power for an AC circuit is: -

\$V_{RMS}\times I_{RMS}\times P.F.\$ where PF is power factor and is the cosine of the angle between voltage and current.

Even this formula makes the assumption that voltage and current contain no harmonics i.e. everything is 50Hz (60Hz in other places) sinusoidal. If there are harmonics present (especially in the current) then the true measure of power is instantaneous voltage and current multiplication in real time, followed by averaging: -



Assuming you have voltage and current waveforms digitized at a high rate your micro can multiply the "instantaneous" values and start calculating "real energy" taken by your appliances/load - this should then be used to provide a pulse for (typically) every kWh of energy consumed.

EDIT - adding section 2.5.5 of reference design for OP: -