Did you know that the main ingredient of toothpaste is wet sand? They call it "hydrated silica" on the ingredients list but that's actually what it is - a suspension of very fine grained silica (sand) particles in water. It's an abrasive - it gets into the tiniest cracks and scrapes out left-over food.
A while back I was cleaning my keyboard (a Microsoft Natural Keyboard Pro):
There was a rather unsightly stain on the front part which had arisen through using the keyboard one too many times with hands greasy from cheese sandwiches. It didn't come off with soap and warm water. I thought to myself "I need some sort of fine abrasive, combined with some kind of detergent that is good for removing organic matter - ah, toothpaste!". I scrubbed at the stain with some toothpaste and an old toothbrush and it came right out.