What we see, and what we keep
What's kept
Nothing that could identify you. There is no database, no accounts, and no record of individual visits. The address you're going to, the page you came from, and your browser's description of itself are used to build the response and then thrown away the moment it's sent.
What we pass on
Every response tells your browser not to say where you came from — the technical name for it is Referrer-Policy: no-referrer. The site you land on is told nothing about the page you clicked from, including the fact that you came through here at all.
What sits in front of us, and what it can see
One honest gap in the claim above, outside our control:
Cloudflare sits in front of this site and decrypts every request as it passes through, so it can see the full address of each one, including the destination written into it, under its own privacy policy. The connection onward from Cloudflare to us is encrypted the whole way, and set up to keep no record of its own.
So the accurate version is: no logs are kept here, and Cloudflare is the one step genuinely outside our control. If that distinction matters for what you're using this for, it should inform how you use it.