Okay, so every wallet has its own public and private keys. I will quickly explain this concept.
Public keys are known to everyone, however private keys are.. private.
You use a private key to encrypt a piece of text and anyone can then decrypt it with the public key.
And since only you known the private key, it can basically be seen as your autograph only then it's safer.
Your private key hence is basically your autograph or password to your wallet. And you can simply write the privately key (s) down as a "backup".
Now what you do with an offline wallet is have two pcs, one offline and one online. First, you create a transaction and sign it on the offline pc. You then transfer the signed transaction (or encrypted text file) to the online pc and transmit it over the network. Your private keys have never been online and your transaction is complete
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