A VFD in normal operation takes the input AC, rectifies it to DC, and then uses transistors to chop this DC into variable voltage, variable frequency AC.
By controlling the AC frequency, you control the speed of the motor. However in order to properly drive the motor you must also vary the voltage being supplied to the motor. The 'constant V/Hz' graph that iwire posted is the simplest control method that is used, and in practise is quite effective.
The internal DC supply is called the 'DC rail', and current is circulating between the DC rail, the transistors, and the motor. There is no direct relation between the current circulating to the motor and the current coming in from the AC supply. In fact, if the motor gets tugged along by an 'overhauling load' then it can start to regenerate, and the AC supply current can fall to zero, with current circulating between the motor, the transistors, and the DC rail, and a braking resistor acting to limit the voltage on the DC rail.
Measuring the output voltage is next to impossible without a tool that can deal with the funky nature of the AC output. As petersonra noted, the transistors in the VFD don't supply a smoothly varying AC voltage to the motor. Instead they switch the full DC rail voltage on and off at high frequency (2kHz to 30kHz and above, depending upon the specific VFD; some old drives or very large drives work at the output frequency directly, however). The high frequency 'square' output is modulated by adjusting the relative positive versus negative duration; it is this modulation of the high frequency pulsing that the VFD can control and adjust.
If you stick a volt meter on the output terminals of the VFD, it will quite likely see the full DC rail voltage, or perhaps 1/2 of this value, and totally ignore the modulation values.
The rather interesting thing about all of this is that the motor windings themselves act as a filter, and pretty much don't respond to the high frequency component of the VFD output. If you look at the motor current on a scope, you will see a pretty good sine wave at the modulation frequency, with a little bit of 'wiggle' a the 'carrier' frequency. So the output of the VFD is high voltage, high frequency, but as far as the motor is concerned you are applying low frequency controlled voltage.
-Jon