
This victory condition does not require you to control the map, but it does require researching an enormous amount of tech.
Influence Victories are also well-suited against aggressive players because taking enemy planets denies them that influence while adding to yours. Diplomatic technologies can help with this. As with all the peaceful victories, managing how much war aggressive neighbors declare on you is key. Placing them strategically for the maximum effect is absolutely crucial. Starbases are obvious candidates for spreading influence. You must hold that control for 10 turns once achieved. You must control 76% of the galaxy within your influence borders. This is sometimes called a “cultural victory” and is achieved by influencing other powers in the galaxy and convincing them to see your ways. Remember, if you get better lasers and the other guy gets better laboratories, you're on the clock - he'll outrace you unless you can use those lasers to make him pay before he passes you in technology. If you research warfare techs early, make them work for you either by bullying your way into tribute or favorable treaties, or by outright conquest. It’s even better if you can build up a big war chest, research your fancy new lasers, then mass-upgrade right before launching the offensive. It's perfectly viable to "turtle up" and explode out of your worlds with devastating effect later in the game.
You don't have to go to war early and often to win by conquest. You'll probably always be fighting to keep your people happy to some extent as a warmonger, and it's not the end of the world if they're grumpy, but population growth and wealth can be more important to an aggressive empire than to others. Prototype ships make an enormous difference early on and quickly establishing a mining starbase or two can make all the difference in the space war.
Population growth can be a major factor, so limiting your battle casualties is helpful. Invasion Tactics like Biological Warfare make a huge difference when conquering enemy worlds, but they cost money. In this condition for victory, you kill everything, and you win.