Moon Knight has a super weird back story. Everything about the character is weird tbh.
So... Marc Spector was a mercenary nearly killed out in Egypt before a statue of a god named Khonshu. In his last breath of life, Marc has a vision of Khonshu who offers to revitalize him if he promises to carry out Khonshu's name and image. He proceeds to dawn the silver cloak he finds on the statue of Khonshu, gets revenge on Bushman (the man who nearly killed him the first place) and heads back to the United States to form his new life as millionaire playboy, Steven Grant, with the fortunes he gathered being a mercenary.
Now he is Marc Spector, Moon Knight, and Steven Grant. Marc also slaps on a fake mustache whenever he wants to acquire info from the streets as simple cab driver, Jake Lockley.
These four personas are supposedly symbolic of the four primary phases of the moon.
But living these multiple identities is no easy feat for Marc to juggle and eventually he develops schizophrenia, etc.
In the Fist of Khonshu run (MK's second term as a titular character), Moon Knight becomes inhabited by Khonshu himself and gains not only new gear, but actual superpowers... errr... lunar abilities. Hence the crescent on his chest being replaced by that Ankh.
Hope that explains enough lol.