She got angry at me when I broke a cup.

Literal

She [topic-は] I [subject-が] cup [object-を] broke [time-時], me [at-に] stomach-stood.

時 ('time') used as a temporal marker — '[event]時' = 'when [event] happened' — a standard pattern. The clause 私がカップを割った ('I broke the cup') sits before 時 as a relative clause. The matrix predicate 私に腹を立てた pairs the body-part anger idiom with に as the target of her anger — 'got angry at me.' 割る ('to break, to crack') is the transitive verb for breaking dishes, glass, or other brittle objects; intransitive partner is 割れる ('to crack, to be broken').