“Why don't I hang on to that for you.”
You're traveling to another country with your boyfriend. When you got to the airport, he wasn't able to find his passport. You both got worried, but after searching for a few minutes he found it. You're worried that he'll lose it again, so you offer this suggestion.
Why don't I hang on to that for you.
This is a way to offer to do something:
Why don't I drop by your office at around three thirty.
Why don't I call around and see if there's anyone interested in buying it.
This expression sounds confident and in-control. So it's someone who's in a position of authority, or who's close to the person they're speaking to.
When you pronounce a sentence that uses "Why don't I ___", say it like a statement, not a question.
You can also use the expressions "Why don't we ___" and "Why don't you ___":
To "hang on to" something means to hold or keep it, without losing it. You use this phrase when you think that someone might lose something:
If you find a good man, hang on to him.
You should hang on to the warranty, and contact the manufacturer if there are any problems with it.