The main advantage comes with phishing resistance. Standard MFA (time based codes) is not phishing resistant. Users can be social engineered into giving up a password and MFA token. Other MFA types, such as pop up notifications, are susceptible to MFA fatigue. Similar to YubiKeys, Passkeys implement a phishing resistant MFA by storing an encryption key, along with requiring a biometric. The benefit here is that these are far easier for the average user, and the user does not need to carry a physical device. Sure, fingerprints could possibly be grabbed with physical presence, but there is far less risk that a users fingerprint is stolen, than a user being social engineered over the phone into giving creds. For most organizations and users, this is far more secure.
Standard MFA (time based codes) is not phishing resistant. Users can be social engineered into giving up a password and MFA token.
So basically this is just idiot-proofing the system. If you aren't the type of person to give your password or MFA token to another person, then passkeys don't really make better security.
At some point, passkeys will be ubiquitous enough they'll be like low friction SSH keys for web authentication (i.e. there will be no shared secrets in the login process).
It has its own pros.
The biggest one is that it uses asymmetric cryptography. This means that the only one that can sign a challenge given by the server is the one that has the private key equivalent of the public key the challenge was used to be encrypted with.
The challenge is sent to the client, in which the client signs the challenge the server sent with their private key and then sent back to the server. Since the server has the public key, the server can verify whether the signature is indeed from the private key owner. This is possible because the private and public key are mathematically linked. This is the reason why it is phish proof. Though I am not sure whether a phisher can just take the challenge, let the victim sign it and then give back the challenge to the server to login. Can anyone confirm that? They are not better than hardware keys (since they are basically software versions of physical hardware keys), but at least better than passwords. At least the breaches will impose much lesser risks when it becomes a standard.