The Gemara in Sanhedrin 92a states that whoever doesn`t leave some bread on the table at the meals` conclusion will not ever see a sign of bracha. The logic seems to be based on the Zohar, that bracha cannot come ?? ???? , that is, from nothing; it must be founded upon something tangible. We find that both Eliyahu Hanavi and Elisha caused the oil flask (and the flour jar) to pour forth continuously to sustain the meritorious women – but they didn´t make it appear out of thin air. (Melachim a, chap 17, Melachim b, chap 4). The bracha had to be founded upon something already existing. So too, one must leave over some bread at the end of the meal, so the bracha will have something to rest upon. This Halacha was codified by the Shulchan Aruch (Orach Chaim 180, 2)

The Mishna Brura there quotes Rashi, who gives another reason for the Halacha: that is, so there will always be something for the poor. There could be a difference between the two reasons: According to the first reason (quoted in the Magen Avraham) it might be enough to just leave over some crumbs, however according to Rashi you might have to leave over something more substantial. (Shaar Hatziyun, ibid)

One shouldn`t, however, bring a new loaf of bread to the table at the conclusion of the meal. That would constitute Avoda Zara (Gemara Sanhedrin, ibid). But if an entire loaf remains at the end of the meal, there is no need to remove it (Taz, Shulchan Aruch, ibid).