Essentially, the online wallet service hold the private keys for you and perform the transaction for you. You only put trust on them and them actually hold the bitcoin for you. In the case if the service goes down, you lose all you bitcoin with it.
Start learning how to generate a paper wallet or purchase a hardware wallet if you are interested to store your bitcoin long term...
This. If you had access to the private keys, you wouldn't need them and would be able to send fee-free. Their money comes by limiting you to their service.