“I'll keep an eye out for it.”

You're a receptionist in an office. An employee at the company is waiting for a package and asked if it had been delivered yet. It hasn't, but you want to let her know that you'll bring it to her when it comes. You say:

I'll keep an eye out for it.

I'll (do something)

"I will ___" is good for telling people about something that you just decided to do. In the example above, the receptionist didn't know about the package before, but now she does. So she's telling her coworker what she will do about it.

keep an eye out for (something)

To "keep an eye out for" something means to look for it. But "keeping an eye out" doesn't mean that you're actively searching for something. It means that you're looking for it while you do something else. For example, if you lost something you can ask a family member:

Hey, I can't find my ring. I've looked everywhere but I can't find it. Can you just keep an eye out for it?