Mining is necessary to create bitcoins, which happens with a coinbase transaction. But they can be transferred from person to person after that without mining. (Mining is still used to confirm those transactions, to ensure they cannot be reversed later; but the people making the transactions aren't responsible for that part of it.)

Think about the following analogy. Regular money is created with a printing press. But you yourself don't need a printing press in order to make transactions; somebody gives you money, and you give it to someone else. Sure, at some time in the past, someone had to print that money, but that isn't your concern.