Category Page that doesn’t show up on main page

Might want to look at The Loop article, and the section on Excluding posts from some category.

It’s a bit confusing (I don’t know PHP). What do I do? Should I put the code from the textarea of ‘Excluding posts…’ in my index.php? Did I read that right?

Yes. Make a backup of any file you change before hand to you can fallback just in case.

I should point out that Custom Query String plugin can do this if you are willing to install a plugin:
https://drunkenmonkey.org/user/archive/custom-query-string-2.6.zip