This idiom or expression like sentence means:

Keep quiet, or keep it a secret, as in:

A. I’m going to tell you something, but mum’s the word!
B. I understand, I’ll keep it to myself.

I’ll keep it a secret, as in:

A. I’m going to tell you something, but please keep it to yourself!
B. Okay. Mum’s the word!

This idiom or expression-like sentence means:

Keep quiet.
Pretend you don't know anything!