Why does Marshall Mathers go by Eminem or slim shady?

Answer 1

"Slim Shady" is a light skinned individual who acts black. Shady/shade refers to an area or a space of partial darkness. The slang is used to describe a person with dark/black skin. Also noted as referring to a white male who acts like a stereotypical black male.

Answer 2

Because almost all rappers go by a different name as an artist. It started as M&M (M-arshall M-athers) when he did battles at the hip hop shop in Detroit at like 15, then changed it to eminem when he was in bassment at like 17-ish till after infinate, than he was on the toilet one day (i am not kidding) and the name slim shady just sorta popped in his head, and that's when he started the "I don't give a f*ck" attitude and started to do more rap battles, then released the slim shady EP, went to LA for the 97 rap Olympics, went on the wake up show, got heared by dre, who sent Jimmy Ivene to find him, and brought dre his demo. The rest, as you say, is history.