The foundational source is the teaching of Rabbi Yochanan and Reish Lakish that while the Temple stood, the altar atoned for a person, but now, one's own table atones for him (Chagigah 27a:1), a teaching cited and expanded by Noam Elimelekh, Sefer Shemot, Mishpatim:11 and Mishnah Berurah 167:30.
This idea is codified halachically: the Shulchan Arukh and Rema rule that one should bring salt to the table before breaking bread because the table is likened to the altar and the eating to a sacrifice (Shulchan Arukh, Orach Chayim 167:5), a comparison the Mishnah Berurah reaffirms by explaining that a person's table, through giving bread to the poor, atones for him just as the altar once did (Mishnah Berurah 167:30).
Related halachot concerning conduct at the table extend this sense of sanctity: one should not remove the bread and cloth until after Birkat Hamazon, and one who leaves no bread on the table never sees a sign of blessing (Shulchan Arukh, Orach Chayim 180:1), with the Mishnah Berurah explaining that blessing does not rest on an empty place and that leaving over food is meant to prepare for a poor guest (Mishnah Berurah 180:1-5).
The Zohar, as rendered in the Sulam, deepens this theme mystically: a table over which words of Torah are spoken is elevated by the Holy One and stands to purify a person from his sins (Sulam on Zohar, Terumah 507), a table properly set draws blessing from above (Sulam on Zohar, Terumah 504), while a table devoid of Torah is called the table of another power, without any share in holiness (Sulam on Zohar, Terumah 506), and a person's table must be approached in a state of bodily cleanliness so that the sustenance prepared there remains wanted by the Holy One rather than drawn toward impurity (Sulam on Zohar, Terumah 522).
The Tanya adds that food and drink enjoyed for the sake of expanding the mind toward divine service, or in fulfillment of the mitzvah of delighting in Shabbat and Yom Tov, becomes absorbed and elevated into holiness rather than remaining bound to the kelipot (Tanya, Part I; Likkutei Amarim 7:3).
An earlier Geonic source likewise treats the set table as a site bound up with atonement, teaching that Vayechulu, though instituted for the Shabbat table, may still be recited even without a full meal, and associates the declaration of forgiveness of sin with the moment it is spoken (Teshuvot HaGeonim (Shaarei Teshuva) 106).
The Beit Yaakov extends the notion of sanctity beyond the table itself to the house as a whole, teaching that the holiness of a home encompasses every aspect of a person's life and that eating and prayer belong within the house rather than the marketplace (Beit Yaakov, Terumah 20, Beit Yaakov, Terumah 28, Beit Yaakov, Terumah 29).
Addressing the practical import of this concept, Tzitz Eliezer (Tzitz Eliezer 10:32) describes the household table as 'the table of the Most High,' corresponding to the altar, and rules that hosting and providing for a guest is regarded as though one had offered a tamid sacrifice.