The term "I started boiling with anger" is used to describe how angry one became. This usually implies that the person's anger had dramatically risen and at this point, he or she would react to the given situation. Boiling symbolizes having taken so much.
It is indeed an idiom: an expression consisting of a combination of words that has a figurative meaning. The figurative meaning is comprehended in regard to a common use of the expression that is separate from the literal meaning or definition of the words of which it is made. An easy way to identify idioms is to ask whether a direct translation of a phrase would result in the correct meaning in another language and culture.