How to launch Electrum after installing it from an AppImage?

Maybe the software is already installed after running the AppImage? Try searching for the software in the applications overview and start it from there, or try running electrum in a terminal.

If the above didn't worked then the AppImage they have provided is for testing the software, not downloading it. You can download the software by the following commands:

If you want snap (The snap version of the app is out of date and may break):

sudo snap install electrum 

If you want to manually compile:

sudo apt-get update
sudo apt-get install python3-pyqt5 libsecp256k1-0 python3-cryptography
wget https://download.electrum.org/4.1.5/Electrum-4.1.5.tar.gz
wget https://download.electrum.org/4.1.5/Electrum-4.1.5.tar.gz.ThomasV.asc
gpg --verify Electrum-4.1.5.tar.gz.ThomasV.asc Electrum-4.1.5.tar.gz
wget https://download.electrum.org/4.1.5/Electrum-4.1.5.tar.gz.sombernight_releasekey.asc
gpg --verify Electrum-4.1.5.tar.gz.sombernight_releasekey.asc Electrum-4.1.5.tar.gz
sudo apt-get install python3-setuptools python3-pip
python3 -m pip install --user Electrum-4.1.5.tar.gz

For more information refer to the Official download instructions for electrum.