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Please let me pay for dinner.
Literal
Dinner-charge [topic-は] me [by-に] pay-[causative-させて]-[polite-ください].
A characteristic polite 'let me do it for you' construction. 払わせてください combines the causative 払わせる ('let/make pay') with the polite request ~てください: 'please cause me to pay' → 'please let me pay.' Common in social settings when offering to cover a bill. Note that the causative-と-request is a polite offer, not a demand: 'allow me the favor of doing X.'