The challenge Logster at Fetch the Flag CTF 2022 involves exploiting a vulnerable website that uses Apache Log4j version 2, which has a known unauthenticated remote code execution (RCE) vulnerability. The attacker first scans the target website to determine its headers and identifies the programming language used, which is Java in this case. The attacker then creates an Express web server to set custom headers and creates an ngrok tunnel to run the server locally and expose it to the outside world. The attacker uses the Log4Shell vulnerability to inject a custom header with a payload that allows them to scan the website again and eventually retrieve the flag by exploiting another vulnerability in the website's LDAP server.