Do I have to save my private keys?
Yes indeed you should back up your wallet, printing out a paper copy of your private keys is a good way to do this because paper can only be accessed physically rather than being hacked via the internet. Obviously it's not safe to store them unencrypted on the cloud or something. The security warnings are there just to make sure you don't do anything stupid with them, to make you double check before potentially giving others access to them.

Bitcoin Core for example allows you to import private keys to restore a backup. If you print them out as QR codes, many wallets allow you to sweep them