It's less complicated than it looks like. The text is just a poorly written mess, full of options (Fedora vs. Ubuntu, repo vs. no repo, stable vs. beta), and they're explaining how to do this through the terminal alone because the interface that you have might be different from what they expect. And because copy-pasting commands is faster.

Can’t I just download a file and install it? I’m on Ubuntu.

Yes, you can! In fact, the instructions include this option; it's under "Installing the app without the Mullvad repository". It's a bad idea though; then you don't get automatic updates.

A better way to do this is to tell your system "I want software from this repository", so each time that they make a new version of the program, yours get updated.

but I have no idea what I’m doing here.

I'll copy-paste their commands to do so, and explain what each does.

sudo curl -fsSLo /usr/share/keyrings/mullvad-keyring.asc https://repository.mullvad.net/deb/mullvad-keyring.asc
echo "deb [signed-by=/usr/share/keyrings/mullvad-keyring.asc arch=$( dpkg --print-architecture )] https://repository.mullvad.net/deb/stable $(lsb_release -cs) main" | sudo tee /etc/apt/sources.list.d/mullvad.list
sudo apt update
sudo apt install mullvad-vpn

The first command boils down to "download this keyring from the internet". The keyring is a necessary file to know if you're actually getting your software from Mullvad instead of PoopySoxHaxxor69. If you wanted, you could do it manually, and then move to the /usr/share/keyrings directory, but... it's more work, come on.

The second command tells your system that you want software from repository.mullvad.net. I don't use Ubuntu but there's probably some GUI to do it for you.

The third command boils down to "hey, Ubuntu, update the list of packages for me".

The fourth one installs the software.