。
She ironed his shirt.
Literal
She [topic-は] his [genitive-の] shirt [to-に] iron [object-を] applied.
アイロンをかける ('to iron, do the ironing') is a fixed verb phrase, literally 'apply the iron.' The shirt being ironed takes に — the destination of the ironing action. かける ('hang, apply, put on') is one of the most polysemous verbs in Japanese, with dozens of specialized senses depending on what's being 'applied' (アイロン, 電話, 鍵, 時間, 魔法, etc.).