You aren’t wrong, the WebUI is stateless, it doesn’t know of any users nor it stores any other context information.

The certificates are required for the UI client to authenticate with the underlying LXD server itself. Much like the SSH authentication is boils down to creating a public/private key pair and the PK is added to your browser(s) and the public key to the server. I believe this is a good walkthrough of the process for anyone starting out.

At work we use Authelia and HAProxy to get around the need to distribute a certificate for each client. At home I simply use Nginx as a reverse proxy to the WebUI with the proxy_ssl_certificate passing a certificate down to it. Here another configuration example of how to use Nginx to pass the certificate.