。
She got angry at what I said.
Literal
She [topic-は] my said-thing [at-に] stomach [object-を] raised.
私の言ったこと is a relative clause modifying nothing — instead, the modifier-less こと at the end makes the clause itself a noun: 'the thing(s) I said.' The に marks this as the target/cause of 腹を立てる (the angering). The structure parallels 私の言ったことに腹を立てる very directly: 'her anger was directed at the thing-I-said.' This kind of relative clause + こと nominalization is one of Japanese's main ways to refer to 'what someone said / did / thought.'