This idiom or expression like sentence means:

Keep quiet.
Keep it a secret.

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

Also, it can be used to mean: I’ll keep it a secret.

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

This idiom or expression has to do with:

Revealing a secret.
Giving away a surprise.

A. The birthday party was supposed to be a surprise. Who spilled the beans?
B. Your wife, boss! With all due respect, boss, you know how women are!
A. Yes I do, and here’s another surprise: You’re fired for spilling the beans about her!

This idiom or expression means:

Secret.

When someone says “Keep it under your hat,” they mean something like:

Do it quietly.
Keep it a secret.
Keep it to yourself.
Don’t tell anybody about it. Etc.

This idiom or expression like phrase means:

A secret that everyone is aware of.
It’s supposed to be a secret, but not really.

The candidate’s infidelity was an open secret among his staffers.

This idiom or expression has to do with:

Not being a secret any more.
Everybody knowing about the issue.

Her money problems have become public and now everybody knows that she was in jail for stealing.

Also:
Out in the open.
Public knowledge.

This idiom or expression means:

Terrible secrets.
Bad things from one’s past that are hidden.

A. I don’t like it when they ask you all those questions in an interview.
B. Why, are you hiding some skeletons in your closet I should know about?

This idiom or expression means:

Guarded.
Closely held secret.

A. Can I trust you with a secret?
B. Sure, I’ll keep it close to the vest.

Also:
Playing cards close to the vest.