You identify yourself to the network as the legitimate owner of an address by signing a transaction with a private key for that address. So using your coins off

You identify yourself to the network as the legitimate owner of an address by signing a transaction with a private key for that address. So using your coins offline involves signing your transaction using a computer that is air-gapped and then manually transferring the signed transaction to the online computer to be pushed to the network. There are user friendly ways of doing this using QR codes and similar things.