Current Transformer (CT) and Potential Transformer (PT) Sensors - How to Measure Electrical Power Usage?
You need to measure the instantaneous voltage and current at lots of little time slices.  Each (voltage x current) is the power over that time slice.  To get the average power, you average all the values from the short time slices over a larger time range.

Usually you want to be able to correctly handle up to about the 100th harmonic.  For 60 Hz power, that means up to 6 kHz, which means at least 12 kHz sample rate.  Fortunately, significantly faster is still easy to do and compute in even a modest microcontroller.

I'd probably sample at 100 kHz because that should still be easily doable.  That means you get 10 &micro;s of computation per sample, which should be more than enough.  For example, that's 700 instruction cycles on a dsPIC33EP series.

The reason they suggest only a single voltage measurement is probably because the voltage isn't going to change much regardless of what the motor is doing.  You can measure one of the phases and just assume that applies to the other phases, with the appropriate phase shifts of course.