It does this with the power of Tasker and the shell in Android (which you can access without root).
In short, it runs netcfg when you connect (or disconnect) to a network, then parses that input until it finds out what kind of connection you’re on. It also finds the SSID of the network (if applicable) and the IP addresses by calling out to http://checkip.dyndns.org/ and parsing that output. It then stores it all and puts it in variables for Minimalistic Text.
You can find the project as wifi.prj.xml (if you’re on your phone now, you should be able to click that link directly and import it) on my GitHub collection of Tasker tasks. I bundled this as a project, so you should be able to just import it.
If you don’t know how to import a project into Tasker, there’s a HOWTO right here: http://almost-a-technocrat.blogspot.com/2013/04/how-to-import-tasker-projects-profiles.html