How did the Companions come to know of the conversation that took place between the Prophet (peace and blessings of Allah be upon him), Jibreel, and the Angel of Death when the Prophet's soul was being taken, and how did this conversation reach us?

This hadith is very weak. It was narrated by Al-Tabarani, and in its chain is Abdullah bin Maimoon Al-Qaddah, who is a denier of hadith and abandoned. It was also narrated by Al-Hakim, and in its chain is Abu Al-Walid Al-Makhzumi, who used to fabricate hadith upon trustworthy narrators, and therefore, it is never permissible to use it as evidence.