I think computers need a card reader (like a credit card reader) to read the card. Or you can use your phone to read it wirelessly via NFC.
One neat thing about systems like this is that the card itself can perform a cryptographic computation that proves its own "ID", without communicating its private key to the connected computer/phone. So even if your computer was compromised, the ID card connected to it still can't be copied. The card is simple enough that there is less attack surface (as compared to an entire computer), so it's much less likely be be hacked, even if it's connected to a hacked device. Though mistakes do happen, since no system is perfect. So if a vulnerability is discovered, new cards might need to be issued.
Granted, an attacker on your computer (controlling it remotely) could just wait until you log in to your bank via smartcard and then quickly pull all your money out... you need a more complex solution to fix that problem (like cryptocurrency hardware wallets use; they have a little screen that shows the proposed transaction, and you have to physically push a button to confirm it, and then it does another cryptographic operation to authorize that particular transaction).
However, the smart card system does prevent an attacker from simply buying a database dump of email addresses, passwords, SSNs, etc. and using that to get into your bank account.
Also, the Estonia system apparently includes keys allowing the manufacturer to perform card operations. How do I know that won't get hijacked?