Since Elementary uses the same package manager as Ubuntu, usually you can follow instructions written for that distribution.
NOTE: Elementary does not come with software-properties-common installed (which is required for add-apt-repository), so here's the steps I followed to install it.
sudo apt install software-properties-common
sudo add-apt-repository ...
The file you downloaded is a script. After downloading it, you must mark it as executable and then run it in Terminal.
Right-click on the .run file, and go to Properties. Here, go to the Permissions tab and click on the first Execute button and then Close.
Open a Terminal window and then drag-and-drop the .run file inside, and press Enter. Type your ...
Try installing the openconnect packages:
sudo apt install network-manager-openconnect network-manager-openconnect-gnome
sudo systemctl restart network-manager.service
The Choose a VPN Connection Type dropdown will now contain two new entries:
Cisco AnyConnect Compatible VPN (openconnect)
Juniper Network Connect (openconnect)
A reboot might ...