After you have purchased cryptocurrencies on an exchange, that exchange will manage your balance.
How do you keep your bitcoins safe?
The bitcoins are yours, but they are in the exchange’s account, the so-called wallet. In the past, exchange wallets in particular have been the victims of attacks. It is therefore advisable to transfer the bitcoin to a personal wallet. You get the unique code of your coins. Save it on multiple devices. These devices should ideally be offline. But please note that if you lose the code, your money is gone too.