After thinking about this a bunch, I think I get it. The chip itself will get the 6V, but since the LEDs are connected to a positive line, that positive line can be a different voltage.
I will try using a voltage regulator on my 12V to make sure the chip is powered with less than 6V, while powering the LEDs with the 12V.
Yep, you got it.
The "input voltage" is the voltage used by the chip's logic, that is normally in the 5v range.
The "output voltage" is the voltage that the output pins can tolerate, it's a bit misleading really because they don't actually "output" any voltage, they just connect the pin to ground.
So you connect the chip's VCC to 5v and the anodes of the LEDs to say 12v, cathodes to the chip.
But NOT 12v to the chip's VCC, that will let the smoke out.
So affectively you need two power supplies, but that's usually easy to do, just use the pre-regulator voltage if it's of a suitable value.