In medicine, bile is a fluid secreted by the liver and used to help breakdown fats. Because it used to be associated with anger, feeling your bile rise is the same as feeling mounting anger.
How did bile get linked to anger? Pre-modern doctors believed there were four basic substances to the human body, called humours, one of which was yellow bile that controlled anger. It doesn't, but when you feel angry, you'll still hear that you have to swallow your bile.