Question: "What is the history behind the tradition of the bread and the wine on the table, to eat and drink after the Eucharist?"
The custom of eating some prosphora and drinking some wine mixed with warm water right after taking communion is called "the zapivka" in the Russian Church. "Zapivka" simply means "washing" or "rinsing," which in this case refers to washing down any remaining particles of the Eucharist.
It is not easy to find much written on the history of this custom, most likely because that is not a question that many people have thought worth exploring. There are many simple actions that we do all of the time in the Church, but unless someone objects to it at some point, or some controversy arises in connection to it, usually we just assume that this is how it is supposed to be.
Most Orthodox sources that mention this practice simply describe it briefly, and state when it is done, but not much else.
In the American context, this is a question that comes up, because the Russian practice is different than what you would typically see in a Greek or Antiochian parish. In those parishes, the laity usually only partake of prosphora after communing (though some parishes and monasteries do something similar to the Russian practice, such as distributing prosphora soaked in wine to the laity).
Interestingly, this is not true for the clergy -- especially for bishops. If you are in a parish with only one priest, he has to consume the remaining Gifts in the chalice, and so he does not do this separately from cleansing the chalice. But when clergy are concelebrating, those who are not going to consume the remaining Gifts, often do partake of wine with warm water, and a piece of prosphora -- and I am told that this is always prepared for bishops.
I am not sure why the practice is different for the laity -- it could be that it simply became common to extend this practice to the laity in Russia. However, I found two sources that stated that this custom goes back to St. John Chrysostom: one modern writer, and one far more ancient, though still many centuries after the time of St. John Chrysostom.
Archpriest Andrei Ukhtomsky, who teaches at the Kiev Theological Academy, states:
"The zapivka after Holy Communion was added during the time of St. John Chrysostom, who, due to a stomach ailment, ate a barley pita on the patriarchal throne. Due to the authority of the saint, such a practice spread further" (A Brief Historical Look at the Liturgy, translated by by Matfey Shaheen, October 29th, 2021 <https://orthochristian.com/130097.html>).
Unfortunately, he does not elaborate on where this tradition is recorded. This might also explain why the custom is more universally followed when it comes to bishops.
However, Nikon of the Black Mountain was a Byzantine monk who wrote a monastic text in the 11th century, called the Tacticon, which became very influential on Russian monasticism, and he wrote:
"It should be known that with respect to collation in church only those who take communion may partake of the collation privately at the sacristy, consisting of a little of the water only or of the wine if it is fitting, and a small piece of blessed bread—but the others not at all—just as the divine [St. John] Chrysostom permitted those partaking of the communion to taste a little something for the sake of a rinse. (Quoted from R. Allison, tr. Black Mountain: Regulations of Nikon of the Black Mountain. In: Byzantine Monastic Foundation Documents: A Complete Translation of the Surviving Founders’ Typika and Testaments. Edited by John Thomas and Angela Constantinides Hero with the assistance of Giles Constable. Washington, DC: Dumbarton Oaks, p. 405).
