|MadSci Network: General Biology|
The short answer is that cats don't purr with their larynx, and they can eat and purr at the same time because purring doesn't have anything to do with the esophagus (which is the tube through which eating and breathing happens). The exact mechanism by which cats purr is a mystery, but one can be fairly certain that they're not using their voice box because they can do things like eat and purr at the same time. It's generally believed that the sound of purring comes from the vibration of a vein in their chest, but the exact location of the vein that might do that is unknown. For a bit more on cat purring, I'll refer you to a couple of previous answers in the MADSCI archives:
Hope that helps!
Rob Campbell, MAD Scientist
Try the links in the MadSci Library for more information on General Biology.