She told me I could use her room.

Literal

She [topic-は] me [to-に], own room [object-を] use is-okay [quotative-と] said.

Reported speech routed through と言った. The reflexive 自分 inside the embedded clause refers back to 彼女 (the subject of 言う), so 自分の部屋 means 'her own room' — not the listener's. Japanese leaves the embedded perspective intact rather than reshaping pronouns the way English indirect speech does. The embedded ~てもいい preserves the original speaker's permission-granting words rather than paraphrasing them.