So then what is the function of exporting a private key from a wallet? Is it to "transfer" funds from one bitcoin client to another without having to send it through the blockchain?
It's an advanced debug feature. It's not recommended for casual use.
Can it be used as a form of backup for your online wallet?
Not recommended. There are lots of keys in the Bitcoin-Qt wallet, and backing them up one by one in this way is going to be very error-prone, especially as there are change addresses that you can't see.
To backup your online wallet you should use the "Backup wallet" menu option, or copy your wallet.dat when Bitcoin-Qt is not running. Save to some external location (such as an USB stick).
A feature to export the wallet in a human-readable format is underway.