LightDM is a Login manager.
You need Xorg for it to work.
sudo pacman -S xorg
chose all when you install it.
LightDM requiers a greeter to work.
Easiest is to install it with a default greeter.
sudo pacman -S lightdm-gtk-greeter
enable the service for it to run at startup.
sudo systemctl enable lightdm.service
This is how it looks like