When it is said
It is a prayer for home, for a husband and wife who have no children and long for them.
They say it together, and say their own names where the prayer asks for them.
Where it comes from
No author known
Who wrote it?
No one knows, and its age is not known either. The Russian prayer books that print it name no author. One gives it under the heading for couples who cannot have children; another calls it the prayer of a married couple for the gift of children.
Azbyka.ru, Prayers for family well-being (in Russian); Molitvoslov.com, Prayer of spouses for the gift of children (in Russian)
The wedding asks it too
At every Orthodox wedding the priest asks God to give the couple “fruit of the womb” and “fair children”. He asks God to bless them as He blessed Joachim and Anna, the parents of Mary, and Zechariah and Elizabeth. Luke’s Gospel says Zechariah and Elizabeth had no child until God gave them John the Baptist in their old age. The Church sings that Joachim and Anna bore the reproach of having none, until God gave them Mary.
G. V. Shann, Book of Needs of the Holy Orthodox Church (1894): the crowning; Isabel F. Hapgood, Service Book of the Holy Orthodox-Catholic Apostolic Church (1906): the Nativity of the Mother of God; Luke 1:7
What the words mean
“Their children’s children”
The prayer asks that the couple see their children’s children. The words come from Psalm 128, a psalm of blessing on the home of one who fears the Lord. The wedding service prays them too. It asks that the couple see their sons’ sons round their table, like young olive trees.
Psalm 128:6; G. V. Shann, Book of Needs of the Holy Orthodox Church (1894): the crowning
The Scripture in it
God blessed them and said to them, “Be fruitful and multiply, and fill the earth and subdue it; rule over the fish of the sea and the birds of the air and every creature that crawls upon the earth.”
… that you may see your children’s children. Peace be upon Israel!
This mystery is profound, but I am speaking about Christ and the church.
Bible text: Berean Standard Bible, public domain.
In Church Slavonic
Услыши нас, Милосердный и Всемогущий Боже, да молением нашим ниспослана будет благодать Твоя.
Буди милостив, Господи, к молитве нашей, вспомни Закон Твой об умножении рода человеческаго и буди милостивым Покровителем, да Твоею помощью сохранится Тобою же установленное.
Ты властною силою Твоею из ничего все сотворил и положил начало всего в мире существующего – сотворил и человека по образу Своему и высокою тайною освятил союз супружества и предуказание тайны единения Христа с Церковью.
Призри, Милосердный, на рабов Твоих сих (имена), союзом супружеским соединенных и умоляющих о Твоей помощи, да будет на них милость Твоя, да будут плодовиты и да увидят они сына сынов своих даже до третьяго и четвертаго рода и до желаемой старости доживут и войдут в Царство Небесное через Господа нашего Иисуса Христа, Которому всякая слава, честь и поклонение подобает со Святым Духом во веки.
From Azbyka.ru, a Russian Orthodox website, in Church Slavonic with some Russian word forms.