To set up, you need to create an account on LogMeIn, download and run their software on the computer you wish to be able to access, and set the software to allow accessing that computer from the Internet.
You can access a computer that you've previously installed LogMeIn on from any computer that has an internet connection through their browser. There is nothing to install on the computer you are accessing from.
Microsoft Remote Desktop is developed for mostly enterprise users for remote control, not to share the screen with a friend. It lacks firewall support, and is much more complicated to setup properly than some other apps, designed for fast and easy sharing.