Read Proverbs 31:10-31 on Sefaria for the surrounding biblical/liturgical context.
The book of Proverbs traditionally is attributed to King Solomon (~10th c. BCE) but is in fact a compilation of wisdom literature edited over centuries. The closing acrostic of Proverbs 31 (verses 10-31) is generally dated to the post-exilic period (5th-3rd c. BCE) on linguistic grounds. The author is anonymous, though Proverbs 31:1 attributes the preceding section (vv. 1-9) to "King Lemuel," instructed by his mother. Some readers connect the Eshet Chayil to Lemuel's mother, who instructs him on what kind of woman to seek.
The custom of singing Eshet Chayil at the Friday-night table is medieval, attributed to Sephardic communities first and adopted widely after the 16th century. The rationale: Friday night is the moment when the woman of the house has prepared everything, candles lit, food cooked, table set, family gathered. Eshet Chayil is the household's public acknowledgment of all that labor.
Kabbalistically, Friday night is also when the Shechinah (the feminine indwelling presence of God) descends to dwell among the people. The same poem that praises the human woman of the house simultaneously praises the divine feminine; the two are reflections of each other.
Modern feminist scholarship has read Eshet Chayil in two opposing ways. One reading: the poem is a patriarchal projection, the wife is praised entirely in terms of how well she serves her husband and household. The other reading: the poem is one of the few biblical texts that describes a woman as an economic agent, she buys real estate (v.16), runs a business (v.18), employs servants (v.15), and engages in commerce (v.24). Far from being domesticated, she is a Bronze-Age CEO.
Many contemporary Jewish families have reframed the song: instead of (or alongside) the husband singing it to the wife, the family sings it together, or each member sings a verse, or it becomes a song of mutual recognition rather than one-way praise. Sephardic and Mizrahi traditions never quite read it as exclusively about the wife, for them it was always also about Torah, Shechinah, and the Sabbath itself.