。
He asked her out on a date.
Literal
She [topic-は] [date-do (please)-デートしてくれと] him [by-に] was-invited.
Several pieces. デートしてくれ is the casual imperative of てくれる ('do for me') — 'go on a date with me.' The と quotative wraps that as the content of the asking: 'asked, saying "go on a date with me."' 誘われた is the passive of 誘う ('invite'), here making 彼女 the subject of being-asked rather than 彼 the inviter; 彼に marks the agent of the passive ('by him'). Japanese consistently uses passive for 'X was invited by Y,' even when English would prefer the active 'Y asked X out.' デート is a loanword tied closely to 1980s+ Japanese romantic vocabulary.