System dependencies
JetBrains Toolbox is distributed as an AppImage (even though the filename lacks the .AppImage extension), so make sure these dependencies are met: libfuse2 libxi6 libxrender1 libxtst6 mesa-utils libfontconfig libgtk-3-bin tar.
(a) Manual install
Download JetBrains Toolbox from the official website, extract it, add the executable permission, and run it -- it will then copy itself to ~/.local/share/JetBrains/Toolbox/bin, create desktop and autorun entries, and edit your .profile to include its shim folder (so you can run pycharm etc. from the terminal).
(Notably, the shims don't include jetbrains-toolbox itself. If you want that to work (maybe you will disable the autorun?), simply symlink it: ln -s "$HOME/.local/share/JetBrains/Toolbox/bin/jetbrains-toolbox" "$HOME/.local/bin/jetbrains-toolbox")
(b) Install script
There is an install script automating the above process (except the AppImage dependencies), which used to be recommended in the official JetBrains documentation. (It's not there anymore, presumably since it's third-party and they didn't have control over it.)
You can check out the script on GitHub, and you can run it by the command below:
curl -fsSL https://raw.githubusercontent.com/nagygergo/jetbrains-toolbox-install/master/jetbrains-toolbox.sh | bash