This is a very good discussion about paper wallets, but you are missing a key point that underlies the whole concept: a wallet, specifically the wallet.dat file, only contains the public and private key pair that identifies the owner and secures his transactions. A paper wallet likewise has on it, usually both fully written out and encoded in a QR-code, those two keys. All of the transactions (which may just be cryptocurrency bought, sold or traded, or smart contracts) are stored in the blockchain. Accessing those or conducting a new transaction requires a computer connected to the internet (even if it is just a cryptocurrency ATM). Nonetheless, a wallet is literally the key (pun intended) to conducting any transactions (specifically adding information) to the blockchain. Anyone can read a blockchain, but only someone with valid keys can add to the blockchain.

Therefore, when the only copy of the keys is stored on a piece of paper (or in an offline hardware device or on an hard drive in a landfill), no one can move funds recorded on the blockchain from one account (identified by the public key) to another account without the corresponding private key held in the wallet.