How do I hook up a 3V device to a 74AHCT125 (Quad Level-Shifter)?
It's not a level shifter. It's a buffer chip that can be used to level shift in one direction only.

It is good for shifting from a low voltage logic signal (3.3v, say) up to a 5V logic signal. It can't shift down from 5V to 3.3V.

Basically it works by having a fixed voltage that it considers to be a logic HIGH, and any voltage above that will be output as 5V. Any voltage below will be output as 0V.  This is the "TTL Logic Level" specification in the datasheet.

So if you have a sensor that emits a 3V logic signal the HIGH logic value will be above its fixed threshold and will trigger a 5V HIGH on the output. 

In contrast "CMOS" logic uses a percentage of the supply voltage to set the thresholds (usually > 0.6 x Vcc), and a 3V signal is less than that (3.3V is about bang on it, so is "iffy"), so won't work (or not reliably).