You say "let me see" when you're trying to remember something, or when you're checking on a computer or a piece of paper for information. For example, when you're calling to make an appointment for a dentist, you may hear:
A: Would it be possible to make an appointment for some time this week?
B: Let me see... we have an opening on Tuesday at noon. Would that work?
The receptionist says "let me see" while he searches the computer for available times.