End of Lakshmana, brother of Bhagavan Sri Ram, in Ramayana happened when the objectives of Sri Ram Avatar on earth were completed. The story is mainly found in the Puranas. Here is a brief account on as to how Lakshman died in Ramayana.
After returning to Ayodhya, Bhagavan Sri Rama ruled his kingdom for several years. There was peace and prosperity on earth. Dharma was flourishing. His sons and the sons of his brothers were made kings in many parts of the large empire that he ruled. During this period, Sita Devi also disappeared from earth when she merged with Bhudevi (Mother Earth).
Yama (Kala), the lord of death in Hinduism, arrived at the palace of Sri Rama and requested an audience with him. Bhagavan Sri Ram received his guest. Now Yama put a condition that no one should interfere with them while they were in discussion. If anyone interferes, that person should be awarded death penalty. Bhagavan Sri Ram gave the word to Yama.
Lakshmana was summoned and asked to guard the door. He was also told about the condition put by Yama.
While the meeting between Yama and Rama was taking place, Sage Durvasa, the hotheaded saint, appeared in Ayodhya and demanded an immediate audience with Sri Ram.
Lakshman tried to explain the situation but Sage Durvasa became angry and threatened to curse the entire city of Ayodhya.
Lakshman would never allow the people of Ayodhya to suffer.
Therefore, the only option left was to interrupt the meeting.
Lakshman opened the door and informed about the arrival of Sage Durvasa.
Bhagavan Sri Ram now had to award death penalty to Lakshman. He mediated on his Guru Sage Vasishta. He advised Rama that abandoning Lakshman is equal to the death penalty.
When Sri Ram told Lakshman about abandonment, the younger brother told his elder brother that he would accept death but not staying away from Sri Ram.
Lakshman then looked at Yama and Sage Durvasa. He realized that the arrival of both them at the same time had other intentions.
Lakshman then read the mind of Bhagavan Sri Ram. He understood that time has come to end Sri Ram avatar on earth.
Lakshman realized that this particular situation was the play of time for Him to disappear from the earth. He readily agreed to the play of Yama or Kala (Time). He then walked into the Sarayu River and took Jal Samadhi.
Lakshmana then took the form of Ananta Sesha and waited for Rama in the milky ocean.