If someone has a bitcoin written out as a long sequence of numbers and letters on a piece of paper, how can that long sequence of numbers and letters be sent to an exchange or a digital wallet?
That "long sequence of numbers and letters" is the private key of the address. Every (non-custodian) digital wallet uses this text to sign you transactions. Without it, you cannot prove that you own the address, because you cannot provide a valid signature to the network.
AFAIK, behind a casascius bitcoin you can find a private key of a p2pkh address (starting with "1"). Example:
KwR5eVh1V678RioffU9fSVFdKBJP3pK7Fp4VqTsxXGRDR3UjxfQ3
You can import such key to a wallet like
electrum.
Steps:
1) New/Restore.
2) Name your wallet and click next.
3) Import Bitcoin addresses or private keys.
4) p2pkh:<HERE_ENTER_YOUR_KEY>
5) Click "Next"
And just like that you'll have successfully transferred your bitcoins digitally. They're ready to be spent.