In plain words
Then Sumitrā went up to Kausalyā: “Sister! What ignorance is this! Nārāyaṇa Himself, in person, was born from your womb as Śrī Rāma. Since He has taken a human form, He must act in keeping with time, karma, cause and duty. So do not grieve that Rāma is going to the forest. He will be able to come back to Ayodhyā very soon,” and so she consoled her. When her son Lakṣmaṇa came and bowed to her, what did Sumitrā teach him? “My son! Do not think, ‘I am going to the forest.’ Ayodhyā without Sītā and Rāma is itself the forest. The forest where Sītā and Rāma are is itself Ayodhyā. Rāma himself is your father, Sītā herself your mother. So serving them with Prema is itself your foremost duty,” she said. In the same way her second son, Śatrughna, went on serving Bharata.
Lakṣmaṇa a portion of Rāma, Śatrughna a portion of Bharata
What is the reason such a close bond arose between Rāma and Lakṣmaṇa, and between Bharata and Śatrughna? Here you must recognise one important point. When Daśaratha performed the Putrakāmeṣṭi yāga, the Yajña Puruṣa appeared and bestowed pāyasam. Then Vasiṣṭha commanded, “Daśaratha! Share this pāyasam out among your three queens.” Daśaratha had equal love for his three queens. He was not partial to any one of them. So he divided the pāyasam into three equal parts, poured them into golden cups, handed them to his queens and said, “Take the ceremonial oil bath, and after you have received the blessings of our guru Vasiṣṭha, eat this pāyasam.”
Kausalyā and Kaikeyī received the vessels very happily, put them away safely in one place and took the oil bath. But Sumitrā felt no joy at all. “Since Kausalyā is the chief queen, the son born to her is the one who may become king. Or Daśaratha, by the word he gave the king of Kekaya, may crown the very son who is to be born to Kaikeyī.