For example, when you buy a stock, say AAPL, how do you get out your AAPL from the exchange?
To this question you can withdraw altcoins just like you can withdraw your bitcoin and that is how you will get it out. There are some wallet that enable some altcoins to be stored in them like erc20. So if you have a coin in eth blockchain you can use erc20 wallets to safe your altcoins. Remember if you don't have direct access to your coin alone, it is still available to everyone including the exchange.