Can someone explain this tutorial for me? and how to run it properly?
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The tutorial installs an alternative display manager--a basic part of the system, and then has you customize and test it.
Given the nickname you gave yourself there's a concern in the comments that if you try to make the changes you might end up with a system you can't repair, and therefore can't use. I share that concern.
A few words to help you understand what the steps are:
Anything you mess up may give you a system that may be difficult or impossible to use from the graphic user interface. That may happen even if you make no mistakes if the spectrumwm program (spectrum window manager?) isn't compatible with the Ubuntu version you are running or the instructions are faulty. In that case you may or may not be able to go back by undoing the changes you made to the ~/.Xsession file using only an Ubuntu terminal.
I've been using Ubuntu for about five years and I'm not the least bit tempted to try the recipe on my system. It really get's into the guts of the operating system.
Nevertheless Free and Open Source software is about freedom for the user of the software and you are free to do whatever you like. It's in that spirit I thought I'd give you a bit of an explanation. If you do decide to try it I would suggest that, before you start, you create another userid and test it to make sure you can use it to do whatever your current userid can do. Also make a backup copy of all of the files you are changing. These steps may help you undo the changes should you have to--but there is no guarantee.