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 ___":

Why don't we exchange business cards?

Why don't you send in your resume and see?