Most likely a large part of why Sith do this is because they feel that they are truly different people then they were before, and do not want to continue using the same names. Throughout the Star Wars films, there is an emphasis on the fact that when someone falls to the Dark Side they become someone different then they were before, from a certain point of view. For example, Obi-Wan says that Luke's father was dead because he fell to the Dark Side, and Darth Vader himself refused to go by Anakin Skywalker anymore; he was simply Darth Vader. Another example would be Ben Solo going by Kylo Ren, even though he wasn't a Sith he still took on a new name when he joined the Dark Side, and then in Rise of Skywalker
He comes back to the Light Side and switches his name back to Ben.
It seems that it's more of a Dark Side thing then specifically a Sith thing. See here for the difference between Sith and Dark Jedi. Most Dark Jedi seem to do this but as we'll see in a moment not all of them do...
Now as for when this started I'm not completely sure. In Legends the farthest back my knowledge goes is Knights of the Old Republic I and II. There are 5 Sith Lords in these games.
Darth Sion: it's unclear whether or not Sion was his birth name
Darth Nihlus: This likely was not his birth name, but it's never stated.
Darth Traya: Traya is likely not her birth name, most of the game she goes by the name
But she lies a lot in the game so it's possible that Traya is her real name and the other is a pseudonym
Darth Malak and Darth Revan: Malak and Revan were their real names. However Kreia theorizes in the second game that Revan never truly 'fell' but that he was the same person as before, and joined the Sith to stop a greater evil. If so that would explain why he stuck with his same name, and Malak likely did the same since he was his apprentice and wanted to emulate him.
So it almost certainly started at the latest when Darth Nihlus, Darth Sion, and Darth Traya fell, likely shortly after or during the Mandalorian War, but it quite possibly started before then.
In canon, all I really know about would be from the movies and every Sith in them takes on a pseudonym, it's possible Darth Maul's name is Maul since it's never stated in the movies and I don't know what it is, but it's unlikely since every other Sith has taken on a pseudonym.