Oxytocin causes the myoepithelial cells, which surround the alveoli (or tiny sacs in which milk is stored and secreted), to contract so that the milk is forced out into the ductules (which act as little filters from the alveoli). The milk is then pushed along to the lactiferous sinuses, which are extensions of the ducts under the areola that store milk. Breastfeeding mothers usually describe this as a tingling or warm feeling or even a cramping in the breast.
Here is a great video showing this process.

