I can't figure out how to deploy an army onto an enemy planet. I'd heard that you're supposed to research troop transports at some point, but I can't find any mention of such a technology anywhere.
So first you need to recruit the army on your planets. Then if you select that planet and go to the armies screen, in the top left of that panel there is a button named "embark all" this will move all your armies to the orbit of the planet where you can then move them like regular ships. Now move them to an enemy planet and you can click the button to land all of your armies. It's advised that you first bomb the planet to destroy fortifications and increase the chance your armies will be able to take over the planet.