Not a historian, but there are a bunch of reasons, many of which also inform the use of noms de plume by authors. Examples:
- George Orwell was born Eric Arthur Blair. When publishing his first full-length work (Down and Out in Paris and London), he worried that it might be a flop. So he told his publisher to use a pseudonym; that way, if the book did flop, it wouldn't hamper future efforts under his own name. Anyhow, the rest is history.
- Often, one's real name is either long, difficult to pronounce, or has specific connotations that the artist wants to avoid. For examples when any or all of those three factors came into play, look at how many Jewish actors, especially in the earlier 1900s (but even today), use "anglicized" names as their stage names: Ed Wynn (Isaiah Edwin Leopold), Edward G. Robinson (Emanuel Goldenberg), Lorne Greene (Lyon Himan "Chaim" Green), Shari Lewis (Sonia Phyllis Hurwitz), heck, even Natalie Portman (Natalie Hershlag). As for musicians in this boat, look no further than Bob Dylan (Robert Allen Zimmerman).
- It can come from a nickname. Gordon Matthew Thomas Sumner was a schoolteacher in England who'd play in jazz bands at night and on weekends. One gig, he wore a black and yellow striped sweater, and someone said he looked like a bee. You now know him as "Sting".
- It can be done to avoid confusion. David Bowie's real name is David Jones, but people confused him with Davey Jones of The Monkees. Katy Perry's real last name is Hudson, but she didn't want to be confused with the actress Kate Hudson.
- Anonymity. Maybe you don't want people to know your real name, lest they stalk you, or lest the police notice that you have a bunch of outstanding warrants....
And the final "why":
- These people are all, when you get down to it, selling something. What will help them sell more of what they're selling? A catchy one-word stage name like Madonna (although that is her real first name)? Something shorter and easier to pronounce? A name that catches people's attentions? (For example, I find "Katy Perry", with the "-ee" sounds at the end, has a kind of childlike quality that makes the name stand out a bit when I hear it on the radio, say.) If it helps you sell the product, it's A Good Thing.
Anyhow. Lots of whys. When? Certainly 1900, if not well before; if not for musicians, then for many other artists, like Molière (Jean-Baptiste Poquelin, 1622-1673). Commonplace? Well, it was relatively common through most of the 1900s, and really took off when rap/hip-hop started (seems like there's hardly anyone out there rapping under their birth-certificate name).