Basically, it would allow websites to only serve users who comply with website requirements (i.e., no extensions, no ad blockers, only Chrome-based, whatever) whatever these requirements are.
You (your browser) go to a website, example.com, which requires attestation. So you must go to an attestation server and attest your device/browser combo (by telling the attestation server whatever information it requires). If the attestation server thinks you are trustworthy, it gives you an integrity token that you pass to example.com, and then you can see example.com. The website knows which attestation server issued your integrity token, so you can’t create your own.
So no extra software means no attestation server would attest you; means you can’t see example.com. End of story. It’s the same as the current “your browser is not supported” window, only you can’t get around it by changing the user agent.
As usual with these initiatives, bullshit is spread across different specs - this spec by itself implies that any number of attestation servers can exist, and they can check whatever they want, and no browser should be excluded, etc., etc., but practical implementation would probably check installed extensions, etc.
That is conveniently left out of the speck. Attestation server may require signed binary on a client system, it may require whatever it wants really, because why not? It’s a website who decides to trust attestation server or not.