How do you capture a homeworld? I realized my allies are using pressurized water reactors and nuclear thermal engines while I run around with internal confinement fusion drives so I think I found a new business opportunity.
My typical method is to use orbital strikes to get their ground forces to a level mine can deal with, then invade. Typically, they either surrender at about the point where they run out of ground forces from my bombardment, or I roll over whatever they have left. This is the ideal if you're trying to take everything >mostly< intact, as it'll leave most of their installations and population alone; if you don't care about population, you can drop high-rad bombs on them for a few months and wait for them to all die/surrender, then ship their structures and minerals off to your colonies and drop a few automines off.
If you're asking about the physical mechanics of invading, make a transport with drop pods and troop bays, load troops into the bays, take it to a (secured) system with an enemy world, transfer the troops to the pods, and combat drop them on the planet. Once they have boots on the ground, it'll automatically create a colony for you on the planet. Go into that colony's GU tab and you'll be able to attack their ground forces through the normal dropdown bar.
But basically, once they take enough damage to their population and troops on a world, they'll surrender, turning it into the lowest sort of territory. Over time it'll eventually become a normal imperial world and take production levels back to normal. On surrender, you get all of their tech that is better than yours, everything on the world, any shipyards orbiting it, and some of their ships will surrender to you. Note that other colonies of theirs and some ships may not surrender.