The advantage - from my very incomplete understanding - is that your passkeys cannot be phished or stolen from you. So only you from your device can log-in to the site. Which leaves me with the question, how cross-device passkeys work.

There are lots of advantages:

• No need to worry about password encoding, like this emoji debacle for example. Actually there's no need to worry about passwords in general anymore, no more worries about lenghts, encoding, character space, remembering them etc.
• It eliminates that scam where attackers set up a site on a domain that looks like the correct one, because the domain is part of the protocol.
• It eliminates phishing for 2FA because login only works on your device anyway and there's nothing you can be tricked into giving away to an attacker.
• If attackers break into a site and steal the public keys they can't use them for anything.
• Since the whole process is automated between servers and browsers and also standardized, it can be upgraded seamlessly and continously, you can upgrade the protocol, the key lengths, the encryption cyphers etc. with zero impact for the user. New upgraded versions can be distributed to both servers and browsers and they'll just use the highest version they both have.
• 2FA is a core part of the protocol, but again in a way that eliminates phishing: it's basically a way to unlock access temporarily to one specific key in your key vault. You can use a master password, or an USB key, or TOTP codes, or biometrics (fingerprint or face) etc., but NOT cellular texts (SMS) anymore because the vault stays on your devices, no need for another party to send you anything.
• Syncing your vault online and over multiple devices, as well as backup, are also a core part of the approach and will eliminate the worry that you drop your phone and you're screwed forever.

The downside is that there's been a whole bunch of tools and apps and services built around passwords for decades and converting all that mass to passkey tools will take a bit.

There are some other tradeoffs like, right now for example I can reasonably print all my passwords and TOTP codes on a few sheets of paper and achieve an "offline" backup in case of untimely death and so on, it's going to be a bit more cumbersome with passkeys. But I expect there will be ways to optimize that as the technology evolves.

Passkeys, under the hood, use a way of proving your identity that doesn't require you to actually send your password, and also doesn't require you to send your username either.
Because of how it's implemented, the system managing the passkeys also gets to authenticate that the website is who it says it is.

So no private data actually gets sent anywhere, but you can prove your identity while also checking the identity of the site you're talking to, like the SSL lock icon but automated. It's often implemented such that the device that holds they keys can't actually have them stolen from it, and it's integrated with a biometric sensor.
This means it's possible to have a high degree of confidence that the person logging in is physically the same person who created the credential, and not just someone who had their password stolen.

The final perk, is that if you're using something like a phone with a fingerprint scanner, passkeys work as two factors of authentication, despite only feeling like one.
Because the phone verifies your identity via fingerprint (something you are), it can then unlock the key that is uniquely available to the phone (something you have).

Combine that with being generally easier to use, and it's pretty clear why most security experts are pushing them. Security that users will use is better than security they won't, and finally we have easier to use security that's also better than the more difficult options.

Might be too late for that, but BOY do I have a bridge to sell you!