Bitcoin is safe because a very big number of people over the world have a copy of the blockchain, where all the transactions are written down.
Although a transaction is sent to the network almost instantly, it takes time until the transaction gets to be included in a block so it can get into the blockchain (it's called "to get confirmed").
The period from sending it to the network to the point it gets confirmed varies, if the fee is big enough it is indeed an average of 10 minutes. The wait is needed for the safety of the receiver of the money, because if the transaction is not confirmed, an evil sender has options to send the same money somewhere else (double spend) and the initial receiver may get nothing.
TL;DR; You have to wait at least one confirmation (10 minutes or more!) for the safety of the one that receives the money.