• You need to verify transactions aren't fraudulent
  • source of trust on this can't be centralized, thats what visa/mastercard are
• You need to be able to calculate a balance
  • traditional currency does this by physically possessing things
    • you can't "own" data; it's fungible i.e. it can be copied trivially. If I copy my wallet onto your computer, who owns it?
  • digital currency gets around this with a ledger
    • that's the detailed log of transactions
    • the transactions in the log are signed with cryptographic keys, the validity of the signature can be verified with the corresponding public keys. The owner of the private key (which never leaves your PC) is the owner of that wallet and is the only one who can generate the signature - the wallet has a 1:1 mapping to the public and private keys

(This is also why you don't need accounts, your PC generates a random Private key, generates the corresponding Public key and that's your wallet. Everyone accepts that it is your wallet because you have its private key)

This is just the very basic system that is traceable and there are newer schemes that are better.

Edit:
RingCT
Stealth Addresses
ring signatures
Transactions over Tor/I2P
Dandelion++