To protect their head.
This is a rather short answer. as it begs another question : Why only one player has the need to protect his head?
---- The cap (called a scrumcap) is normally worn by forwards specifically front rows and second rows to prtoect them during scrums. It prevents cauliflower ear (a common Wrestling injury) and just general pushing and squishing and pulling on the head and ear. It doesn't protect against concussion or other head and neck injuries much.
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History has not recorded that information although it is known that a team in the 1860s called the Brooklyn Excelsiors wore caps that looked somewhat like today's caps in that they had a rounded top. There were no rules back then that stated players had to wear a cap so most did not wear one although some players would wear a straw hat with a large brim to shade their face from the sun.
You can, but most players don't. If the helmet is too big, wearing your cap under it is a good way to make it fit better though.
The cap colouration/symbol designate what unit they're from.
Bessie Coleman usually wore long pants with a white t-shirt (her airplane wear) When she went flying, she wore a cap and goggles.
He is depicted wearing a Chicago White Sox cap on his Hall of Fame plaque.