。
Please let me take a day off tomorrow.
Literal
Tomorrow 1-day rest-let [request-てください].
休ませて is the causative te-form of 休む ('rest') — 'make/let rest.' Combined with ~てください, it becomes the 'let me do X' construction: literally 'please make me rest,' meaning 'please allow me to rest.' This causative-request pattern is the standard way to ask permission from a superior to take time off or do something on one's own.