"Let me know" means "tell me". But "let me know" is a more polite and friendly expression. It's very common to hear in an English-speaking workplace.

If anything comes up between now and then, just let me know.

Think it over and let me know in the next week or so.

"Let me know if you have any questions" is a common way to politely offer to answer someone's questions about something. It's also a good way to end a business email or telephone call when you've given someone information and don't need to ask them for anything. For example:

Hi Fran,

Attached is a summary of our costs for this month. Let me know if you have any questions.

Thanks,

Aaron