How does "Sign Message" work? How does "Sign Message" option in the Bitcoin QT client work? I can't find any description or documentation of it.

Do I have to use this option to sign some message when I receive bitcoins? If so, how do I do that and what should I write?

This is how it looks (Sorry for danish client language):
You don't have to manually verify Bitcoins you receive - the according transaction is verified automatically by your Bitcoin client. If you see a green checkmark left to the transaction you received (in the Transactions view) then it means that the rest of the Bitcoin network has confirmed the payment as well and you can safely consider the coins yours.

The "Sign Message" feature is an advanced functionality of the Bitcoin client which allows you to sign arbitrary messages to prove to somebody that you are (were) in control of the funds of some Bitcoin address. For a usage example see this answer. You normally don't need this feature for day-to-day use of Bitcoin.