Unfortunately I do not understand the situation better with your latest response, so I’ll try to explain from the other side what does this do, etc.
Normally, when you enable a site to work as HTTPS, Cloudflare just proxies the site over a secure connection (to Cloudflare, Cloudflare presenting their own TLS certificate to the users of the site). This works if you go to https://boyan.press and your domain goes through Cloudflare, and the DNS record in Cloduflare’s configuration is set to proxy all requests (orange cloud next to the relevant DNS record).
The way this works is by Cloudflare serving in the DNS response their IP, instead of your servers’ IP, and when someone browses to your site, Cloudflare on the background connects to your server, either securely or not (latter option available only if your SSL mode is set to “Flexible”), and whatever it gets in response to the background request, it returns to the originating client.
So what does “Automatic HTTPS Rewrites” do? Well, sometimes people are unable, or unwilling, to make sure that all the references to other files (images, CSS, JS) within their site are either relative to the current path (like …/images/file.png or /file.png), so they’ll load at the same protocol as the whole page was loaded (HTTP if the page was HTTP, and HTTPS if it was HTTPS), and they have internal links that start with “HTTP” even if the page is loaded as “HTTPS”. When that happens, the page is said to have “mixed content”, which is a security issue, and as such will display with a broken padlock. There are two solutions to that issue - either fixing the links to be either relative or explicitly begin with “https://”, OR, for those who can’t or can’t be bothered, Cloudflare has the option of “Automatic HTTPS Rewrites”, which basically means that once your HTML pages pass through Cloudflare’s network, they will, on-the-fly, make those corrections in the URLs, and convert “http://” to “https://” for any URL that is under your domain. For other domains they can’t do that because they have no idea if the other domains support SSL (yours, they do know, after all, they’re serving your domain through SSL…). So for any remote content, you should still have Mixed-Content even with this on.
Now to your site:
Your domain, assuming it is “boyan.press” like you say, does not even have Cloudflare’s servers set up on the domain WHOIS record. The DNS servers for this domain right now are DNS1.REGISTRAR-SERVERS.COM and DNS2.REGISTRAR-SERVERS.COM. This means that there’s NO WAY that any of your traffic passes through Cloudflare, because the very first prerequisite - having Cloudflare control the traffic by pointing the nameservers for the domain to them, is not set up. You were told to set it up when you added the domain to Cloudflare (and probably also in a follow-up e-mail).
Since your traffic does not fly through Cloudflare network, it doesn’t matter what change you’ll configure in Cloudflare’s dashboard - Cloudflare cannot affect your traffic until it is pointed to them; As such, the change you made (“Automatic HTTPS Rewrites”) had no effect on anything.