You use signing to prove to someone else that you are the owner of a bitcoin address.

The most common purpose to do so is when you pay something to someone else: You use the address from which you sent money to sign a message containing your purpose of the payment. This way, the recipient knows who paid him and nobody else can claim that he made the payment.

For example, a signed message could look like this for such a purpose:
message: "i am forum member xyz. as agreed, i sent you btc0.1 for exchange to USD. please send the USD 75 to bank account ..."
signing address: ...
signature: ...

Signing the purpose of a payment with the sender address is important, because everybody can see your transaction to the recipient (transactions are public) and if someone else knows the recipient, this person could tell the recipient a made-up story why he paid this to him and claim the return (goods, other currency, ...) for the transaction.