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 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:
5) Click "Next"
And just like that you'll have successfully transferred your bitcoins digitally. They're ready to be spent.