Meaning

noun top part of an apron; covering the chest

verb drink moderately but regularly

noun a napkin tied under the chin of a child while eating

originOriginally verb sense “drink heartily”, from Middle English bibben, either from Latin bibō (“to drink”) or of imitative origin. Noun sense (clothing) presumably either because worn while drinking, or because the clothing itself “drinks up” spills.

Etymology from Wiktionary, CC BY-SA 3.0