Let me add some more information after @Ayushya's answer. This is of course speculation as I don't happen to run any major exchange myself, but there aren't very many options on how this can work really.
Exchanges do as much as possible with internal bookkeeping. Whatever buys & sells you do happen only inside their own centralized database and no actual cryptocurrency is transferred anywhere. This way they save on transaction costs and speed up the process.
When you want to transfer something to your exchange "wallet" you send it to a wallet which is generated by the exchange for you. So the exchange basically generates a private key and checks its corresponding public address. Whenever the exchange notices that the wallet receives assets it uses the wallet's private key to transfer the assets forward into its internal hot wallets for safekeeping. This transfer happens in the blockchain.
When you want to transfer assets out of the exchange the process is simpler - you just receive the assets from the how wallets.