Curl isn't the problem they're describing. The remote can assume you're running it in a terminal (especially since the user agent string indicates you're using curl) and can send malicious escape sequences in the body, which will be interpreted by your terminal emulator in most cases.
This is true of any program that prints output directly from a remote host / untrusted source.
In the event your terminal emulator has a vulnerability or allows you to run arbitrary commands (this is a feature of some emulators), the site can target that functionality for users of that emulator and wreak havoc.
I maintain a lot of ANSI escape related code. These exploits have always been theorized, but I've not once heard about this being exploited in the wild. It's certainly possible. Not very probable. Refer to your threat model, as always.