How does a TLC5917 LED Driver work?
If you look on the datasheet you'll see that the external voltage applied to the LEDs is on the anode of the LED. The cathode of the LED is connected to the pin of the TLC device. 



In this case the TLC is sinking current. To reiterate, the driver doesn't supply the voltage, the voltage is applied externally. The driver does, however, ensure the correct current flows through the LEDs. 

It should also be noted that the driver takes up any difference between supply voltage and what is dropped across the LED. So you have to take care that you don't have a combination of external voltage and current drawn that would damage the chip. The higher VLED the higher the power dissipated by the the TLC. 

The image below gives you a very basic internal schematic of the IC.



Remember that the LEDs' cathodes are connected to OUT0, OUT1 etc, with the anode connected to the external voltage. The little circles with the down pointing arrows are representing constant current sink. It is this that ensures the current drawn by the LEDs is the same and constant. 

It might be that you're not familiar with the difference between a current source and a current sink, so it might be worth doing a little reading up of the difference between them if that is the case.