The Begging-bowl refers to the the story that shortly before the Buddha reached enlightenment, a young woman named Sujata offered him a bowl of milk-rice. At that moment, he was practicing austerity by eating extremely little. But he realised at that moment that he would need to have more strength for the final steps to enlightenment, and further fasting would only reduce his energy. After he reached enlightenment, he is said to have thrown away what little was left in the bowl to signify his renunciation of all material possessions. Finding the middle way between extreme austerity and complete attachment to life is an important principle of Buddhism.
The bowl also points to the monk's way of life; going from the monastery into the village each morning and living off what is put into it by lay people.