For an example of how cold storage works, let's say that you have a hardware wallet. To use it, you first connect it to your computer. Next, you choose the option to receive crypto, which will generate an address. You can send your crypto to this address to store it on your cold wallet. The device has both your public and private keys, so once you disconnect it from your computer, that information is completely offline.
If you want to transfer crypto from your cold wallet to another address, it's a similar process. Connect the cold wallet to your computer, enter the address, and send your crypto.
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