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 ___":