。
She forgave the boy for his rudeness.
Literal
She [topic-は] boy [subject-が] rude [nominalizer-の] [object-を] forgave-for-him.
The nominalized clause 少年が無作法なの ('the boy's being rude') becomes the object of 許す (ゆるす, 'to forgive, to permit'). The ~てやった form uses the benefactive やる (casual version of あげる), meaning she did the forgiving as a favor — forgiving him generously from a position of authority or seniority. 無作法 (ぶさほう, 'rude, ill-mannered') is a formal word for bad manners.