Step 1: Downloading the archive
Download the latest MCreator Minecraft mod maker version for Linux and extract the tar.gz archive to the desired location of your choice.
Step 2: Launching the software
Go into the MCreator folder you just extracted and open it in the terminal.
Once you open the terminal in the folder of MCreator, run the following command in it:
./mcreator.sh
The command should look like this in the terminal:
Step 3: Desktop shortcut
Once you launch MCreator with this command, it will make a launcher icon in the installation directory and on the desktop (on supported Linux distributions). You can use these icons to launch MCreator without opening the terminal each time. Keep in mind that if you move the installation directory, you need to do Step 2 again.
If you are looking for installation instructions for older releases, check MCreator's Wiki for the legacy installation procedure.