She wanted to be alone.

Literal

She [topic-は] alone be-want-was.

いる ('to be, to exist' for animate beings) plus the auxiliary ~たい ('want to') gives いたい ('want to be'); past form いたかった expresses the wish in past tense — 'she wanted to be.' Reporting another person's want with bare ~たい is mildly odd in modern Japanese (it usually requires ~たがる, '~want-shows,' or quoted/embedded speech), but in narrative or third-person prose ~たかった with a past-tense feel is widely accepted. The te-form ひとりで anchors the desired state.