Make a site like MXtoolbox/dnsgoodies

Well, how does a 'submit' button on a forum post page interact with the web server to run the required script and then display the output (post) to the webpage?

There are a number of scripts and pre-built modules that can help you do what you want, or you can create your own from scratch. Either way, you need to learn web programming. PHP is a pretty standard way, but other languages can be used as well, like perl, ruby, python, java, and the list goes on. Not to mention what kind of web server you want to deploy on, let alone which OS…

You have much to learn, and the journey starts with google…

Note that a web site that allows people to run scripts or make outgoing connections on the basis of arbitrary user input is a security exploit waiting to happen. Be sure you know what you're doing before you proceed.