Ok, I don't understand how bitcoin works, but ultimately they're just encryped hash files on a disk, right? So unless the other person spends them before you do and you have a backup, how can they be stolen?
Someone please explain...
Loosely speaking, a Bitcoin is a secret number. To spend a Bitcoin, you send that number to someone else and the transaction is recorded by the Bitcoin network. The network keeps track of who owns each coin, which means others can verify the ownership of a coin by consulting the network.
The exchanges work by you "spending" your Bitcoin to them, so now in the network's eyes, the exchange is the owner of that coin. In return, the exchange keeps a record that you have a certain number of Bitcoins in your account. The idea being that in the future you can instruct the exchange to send coins back to you, to other people, or to deduct an amount of coin and send you the cash instead.
It's just like a bank; you hand a teller $20, the bank adds $20 to your balance, and they keep the $20 bill.
Where these exchanges differ from a bank is in their lack of accounting ability, apparently.