Preparation for Birkat HaMazon
The Torah, after expanding on the praises of the Land of Israel, orders us: פAnd you shall eat and be satisfied, and you will bless HaShem your God for the good land which He gave you.צ (Devarim 8:10.) This is the Torah commandment to say the birkhat hamazon, the grace after meals.
Ideally, we should not just rush into this blessing immediately after our last bite; rather, the halakha records many laws and customs which create the proper atmosphere for this most important mitzva.
WASHING HANDS AFTER MEALS - MAYIM ACHARONIM
There is an obligation to wash the hands after the meal. The Talmud explains that the hands tend to be dirty after a meal, and in particular may be encrusted with salt which is dangerous if rubbed in the eyes. And blessing in sanctity requires clean hands, a theme we have seen many times. (Berakhot 53b.)
The water from this washing should not create a puddle where someone might step, but rather be collected in a vessel or absorbed into the floor. The Talmud explains that an פunclean spiritצ dwells on this water. (Chullin 105b.)
We have previously explained an פunclean spiritצ or spiritual defilement as a lost or frustrated potential for sanctity. This explanation is appropriate here as well. Food eaten in the proper spirit of sanctity is elevated as it is made part of the body, the abode of the divine soul. But what is left over on the hands has lost its potential for this elevation to holiness. Until the hands are washed, there is still the chance that the food sticking to them will be eaten, but washing the hands concludes the meal and eliminates the chance of the rinsed-off remainders to be assimilated to sanctity.
When washing prior to a meal, it is proper to raise the hands, showing that the ultimate source of material blessing is in the higher worlds of the spiritual. (SA OC 162, see Beֶer Heitev 2.) When washing subsequent to a meal, it is proper to lower the hands (SA OC 181:5), showing that materiality which is not assimilated to sanctity belongs to the lower worlds, those which are separated from holiness.
Leaving Bread on the Table
Bread is left on the table to symbolize that G-d provides us with more than our needs, as well as to provide a foundation for further blessing. (SA OC 180:1.) The Mishna Berura gives the example of the miracle performed by Elisha, in which a single cruse of oil was able to fill scores of other vessels (Melakhim II beginning of ch.4).
This reminds us that the concept of blessing - in Hebrew פberakhaצ - relates specifically to the growth and development of something existing - not to the creation of something new.
The Zohar likens the bread left on the table to the show-bread left at all times in the Mikdash. The table of the showbread was likewise meant to be an anchor and a focus for the earthly blessings bestowed on the Jewish people (Zohar Trumot, II:157b, mentioned in Mishna Berura s.k. 4). The idea that the shulchan is a source of wealth is hinted at in the gemara, which says that one who wants to be enriched should tend in his prayers to the north - the direction of the shulchan in the Mikdash (Bava Batra 25b, see Rema OC 94b).
A Knife on the Table</p>
There is a custom to cover any knives on the table during the benching - except on Shabbat. (SA OC 180:5.) One reason given (see MB) is that the table is likened to the altar, which comes to lengthen life and therefore is built without iron implements which shorten life. (Rashi Shemot 20:22.)
Of course, Jews are not forbidden from using iron implements in general, and we use them even in the Temple itself (to slaughter the sacrifices), just as we may use our knives on the table during the meal. Part of manֶs Divine image is his creative ability, which reflects HaShemֶs ability to create פex nihiloצ - something from nothing. The mirror image of this capacity is our destructive ability. Both are symbolized by iron, which enables us to hew stones into carefully planned shapes or on the contrary to commit violence in carefully planned ways.
However, in order to sanctify our creative side it has to be set upon the proper foundation - a foundation of submission to G-d and an acknowledgment that ultimately He is the Creator; we are at most His agents. Creation without this recognition is itself destruction - it is a sacrifice to the pagan god of our own talents, saying פMy strength and the might of my hand made me all this wealthצ. (Devarim 8:17)
We do indeed slaughter sacrifices in the elaborately designed Temple, but when we want to elevate them before God we must start with an altar built with unhewn stones, left exactly the way G-d gave them to us. We use our knives throughout the meal, but when we want to elevate our eating, by affirming that the Source of our material blessing is in G-d, it is appropriate to cover them. This brings us back to the point we made above, that the idea of blessing is that of augmentation and not of creation.
On Shabbat, our creative capacity goes on vacation entirely. All creative labor is forbidden. Then we are not concerned with the symbolic connection of metal to our ability to create and destroy, and we need not cover the knives.