。
She said I could listen to the records.
Literal
[topic-は] records [object-を] listen good [quotative-と] said-for-me.
聞いていい is a casual contraction of 聞いてもいい ('it's OK to listen') — the も is dropped in informal speech. The と marks the quoted permission, and 言ってくれた adds the giving auxiliary ~てくれる, framing the speech act as a favor done for the speaker (her granting permission was beneficial to me). This is a classic case of how 授受表現 (giving/receiving) tracks who benefits from an action — English just says 'she said' but Japanese marks the perspective.