As I understand it, instead of the website or online service storing your password (in a, supposedly secured way), with passkey your password manager stores a private key and the online service stores a public key (or rather a lock). The key and the lock are paired together cryptographically (mathematical functions that are non-reversible). Now when you login with passkey, the service sends a challenge generated from the lock, that can be solved only with the matching private key, your password manager solves the challenge and your authenticated. Locks and keys were not exchanged during the process, and services never store your key. Everything happens automagically.
It actually uses the same protocol used is some hardware security keys such as Yubikey and Solokeys. The problem remains the same as with hardware security keys, adoption and support, compatibility. It's very rare that a service supports these options, although they exist for a while.
Anyone feels free to correct me if I wrote something wrong. I am by no mean an expert.
The (over?) simplified version is they’re basically the same as the key/certificate pairs you use to connect to a website securely while also proving its identity to you.
Some key benefits of passkeys are:
Your private key doesn’t leave your device (or your password manager). You no longer have to worry about if the website you’re using is incompetent and storing your password in plain text waiting to be stolen in a breach. The only one who can expose your passkey is you (or your password manager)
Your passkey isn’t something you have to remember so for the unwashed masses it’s more idiot proof because they’re more secure by default