She got angry when she read my letter.

Literal

She [topic-は] my letter [object-を] saw-and, stomach [object-を] raised.

腹を立てる literally means 'to raise one's stomach,' a vivid idiom rooted in the older Japanese concept of the 腹 (hara, belly) as the seat of emotion and true intention — the same notion that gives us 腹黒い (literally 'black-bellied' = malicious) and 腹を割って話す ('split open one's belly' = to speak frankly). Western culture historically located feeling in the heart; classical Japanese located it in the gut. 見る here means 'read' (literally 'see/look at') — using 読む would emphasize the act of reading more deliberately; 見る suggests catching sight of the content.