Here's my abridged version.
They are known as "Good Daedra" because they appeared before the Aldmer pilgrim known as Veloth and taught him New Ways of living. He founded a splinter sect in Aldmer society and led a mass Exodus to a new land later known as Resdayn and later still as Morrowind. Along the way, the three Good Daedra instructed and guided Veloth in how to properly arrange this new society, the Chimer.
The followers of Trinimac appeared and tried to stop the Exodus, but Boethiah tricked the god-king of this sect and ate him. He defecated the remains out as Malacath and the followers of Trinimac became the Orcs.
Finally the Chimer arrived in their home land. There's some other details about their dealings with Dwemer and Nords but that's not important when discussing the Good Daedra. In short, Boethiah, Mephala and Azura each taught the Chimer priests something useful about forming society. They taught them secrets of murder and intrigue, and how to build Houses and live in them (more like a clan or a tribe than a literal house.) among other secrets.
Eventually, the War of the First Council happened and three advisers to the Hortator or Warchief of the Chimer betrayed him and used a profane artifact to make themselves gods, effectively taking the place of the three Good Daedra in Chimeri worship. They claimed the Daedra had been their "Anticipations." Sotha Sil took Azura's spot, whereas Vivec took over Mephala's place and Almalexia took Boethia's. Azura became angry and cursed the Chimer, who then became Dunmer and whose skin and eyes changed.