The context of the hadith is that under Islamic law, consuming human milk causes a foster relationship to be established between the consumer and producer of the milk. It means that a wet nurse is considered equal to the biological mother of the person who drinks her milk. Subsequently, most of the mahram pairs that are mentioned under Consanguinity and Affinity in " Who are the women that a man is forbidden to marry? " also extend to milk kinship.
Hence if woman A produces milk, and this milk is consumed by woman B, then B is considered the daughter of A. The husband of A can never marry or make a concubine of B since she will be his wife's daughter and hence a mahram who is forever forbidden to him.
The athar quoted in the question is about an incident where a woman became jealous of her husband's concubine. So she fed her with her own milk, in order to make it illegal for her husband to have sexual relations with the concubine.
The woman might have been under the impression that drinking milk caused kinship to be established unconditionally, even if the consumer was already an adult. However Umar ibn al-Khatab ruled against her plot and clarified that kinship is only established when milk is consumed by an infant. Drinking of milk by an adult has no legal effect.
This judgement is actually the madhab of the vast majority, that kinship is only established when the milk is drunk by an infant, close to the age of 2 years. The view is based on the following evidence from the Quran and Ahadith:
والوالدات يرضعن أولادهن حولين كاملين لمن أراد أن يتم الرضاعة
Mothers may breastfeed their children two complete years for whoever wishes to complete the nursing [period].
— Quran 2:233
وفصاله في عامين
weaning is in two years.
— Quran 31:14
فإنما الرضاعة من المجاعة
Suckling relationship is established only when milk is the only food of the child
لا رضاع إلا في الحولين
Suckling applies only (to infants) during the first two years
— Bulugh al-Maram
لا يحرم من الرضاعة إلا ما فتق الأمعاء في الثدى وكان قبل الفطام
No prohibition results from suckling except for what penetrates the intestines and is taken before time of weaning.
— Jami at-Tirmidhi
Note that while this is the majority madhab, a diverging opinion also exists which accepts the legal effects of any Rada' that occurs in adulthood.