The Shulhan Aruch, in Siman 180, rules that bread must be left on the table during Birkat Hamazon. One explanation is that Birkat Hamazon draws down the Beracha from Heaven, and there is a general principle that blessings do not descend upon empty vessels. This is derived from the story of Elisha and the wife of Ovadia, in which he told her to use her last remaining oil to perform the miracle. Leaving bread on the table serves as the basis for the continued flow of Beracha from Hashem. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) went one step further that one should leave all the remaining bones, shells and peels from the meal. Apparently, the Beracha of Birkat Hamazon can extract from them sparks of holiness. Of course, one should collect them in an orderly fashion.
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One should be very careful not to step on remaining pieces of bread or show disdain for them. Doing so leads to poverty. If the pieces are less than a Kezayit, they may be discarded respectfully. If there are bigger pieces, which will not be eaten, the Poskim allow sealing them in a bag and then placing the bag in the garbage. The precedent for this is discarding Teruma nowadays and fruits with Kedushat Shevi’it. This Halacha would also apply to cake and other Mezonot, as well.
SUMMARY
One should leave bread on the table during Birkat Hamazon.
Stepping on bread leads to poverty. Pieces larger than a Kezayit may be sealed in a bag and placed in the garbage,
