。
Mother left ten minutes ago.
Literal
Mother [topic-は] 10 minutes ago [at-に] went-out (polite).
前に here is 'ago,' with に marking the point in time (the moment that is 10 minutes previous to now). 前 without に — 10分前出かけた — isn't ungrammatical but feels more abrupt; with に it locates a specific moment. 出かける specifically means 'to leave (the house/office), go out' and is distinct from 出る (exit generically).