Passwords are based on symmetric cryptography. When you log in to a site using a password you give the site your password in plain text, hopefully over an encrypted communication channel such as HTTPS so that no one between you and the site can see the password.
The site then takes that plain text password and decides if it matches the plain text password you gave them when you created the account or most recently changed your password. If the site is following good security practices they aren't actually storing a copy of the password in plain text. They will store a hash of it and compare a hash of the password you just send to see if the hashes match.
Passkeys are based on asymmetric cryptography, also known as public-key cryptography. When you set up an account at a site to use passkeys your device generates a public key and a matching private key. The site is given the public key and your device keeps the private key.
When you want to log in later the site sends your device some data, your device does a computation on that data that involves the private key and sends the result to the site. The site can recognize that whatever did that computation had access to the private key that corresponds to the public key the site has on file.
Passkeys use public-key authentication wherein the server only stores the public half of a keypair and the client authenticates by correctly signing a challenge sent by the server, which the server then verifies using the public key.
At no point is the private key ever sent over the network or otherwise exposed to any infrastructure or code controlled by the server.
Passwords are reasonably secure since we've been using them for a long time but there is in fact a huge chain of trust required to keep them secure and links in that chain frequently break.
That's the benefit you get from passkeys that no password manager will otherwise be able to give you.