Could you give me a lift to the post office?

Literal

Post-office-until ride-on-[causative-せて] give-[honorific-くださる]-[negative-ませんか]?

An ornate polite request using multiple layers. 乗せる ('to let someone ride, give a lift') — the causative of 乗る — is used for giving rides. 乗せてくださいませんか combines the te-form request with the honorific くださる and the polite negative interrogative ~ませんか — literally 'won't you kindly let me ride?' The まで here is 'as far as,' marking the destination.